05-14-2023-1937 - variety continued... (draft)

In the field of psychology, cognitive dissonance is the perception of contradictory information and the mental toll of it. Relevant items of information include a person's actions, feelings, ideas, beliefs, values, and things in the environment. Cognitive dissonance is typically experienced as psychological stress when persons participate in an action that goes against one or more of those things.^[1] According to this theory, when two actions or ideas are not psychologically consistent with each other, people do all in their power to change them until they become consistent.^[1][2] The discomfort is triggered by the person's belief clashing with new information perceived, wherein the individual tries to find a way to resolve the contradiction to reduce their discomfort.^[1][2][3]

In When Prophecy Fails: A Social and Psychological Study of a Modern Group That Predicted the Destruction of the World (1956) and A Theory of Cognitive Dissonance (1957), Leon Festinger proposed that human beings strive for internal psychological consistency to function mentally in the real world.^[1] A person who experiences internal inconsistency tends to become psychologically uncomfortable and is motivated to reduce the cognitive dissonance.^[1][2] They tend to make changes to justify the stressful behavior, either by adding new parts to the cognition causing the psychological dissonance (rationalization) or by avoiding circumstances and contradictory information likely to increase the magnitude of the cognitive dissonance (confirmation bias).^[1][2][3]

Coping with the nuances of contradictory ideas or experiences is mentally stressful. It requires energy and effort to sit with those seemingly opposite things that all seem true. Festinger argued that some people would inevitably resolve the dissonance by blindly believing whatever they wanted to believe.^[4]

^{https://en.wikipedia.org/wiki/Cognitive_dissonance}

^{https://en.wikipedia.org/wiki/Social_class}

^{https://en.wikipedia.org/wiki/Social_theory}

^{https://en.wikipedia.org/wiki/Social_phenomenon}

 

Sociological imagination is a term used in the field of sociology to describe a framework for understanding social reality that places personal experiences within a broader social and historical context.^[1]

It was coined by American sociologist C. Wright Mills in his 1959 book The Sociological Imagination to describe the type of insight offered by the discipline of sociology.^{[2]: 5, 7} Today, the term is used in introductory sociology textbooks to explain the nature of sociology and its relevance in daily life.^[1]

^{https://en.wikipedia.org/wiki/Sociological_imagination}

The imaginary (or social imaginary) is the set of values, institutions, laws, and symbols through which people imagine their social whole. It is common to the members of a particular social group and the corresponding society. The concept of the imaginary has attracted attention in anthropology, sociology, psychoanalysis, philosophy, and media studies. 

^{https://en.wikipedia.org/wiki/Imaginary_(sociology)}

 

^{https://en.wikipedia.org/wiki/Social_group}

^{https://en.wikipedia.org/wiki/Antisocial}

 

^{https://en.wikipedia.org/wiki/Isolation}

^{https://en.wikipedia.org/wiki/Solitary}

 

^{https://en.wikipedia.org/wiki/Unknown}

 

^{https://en.wikipedia.org/wiki/Uncertainty}

 

^{https://en.wikipedia.org/wiki/Stochastic}

 

^{https://en.wikipedia.org/wiki/Assimilation}

^{https://en.wikipedia.org/wiki/Bias}

^{https://en.wikipedia.org/wiki/Observational_error}

 

Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.^[1] In statistics, an error is not necessarily a "mistake". Variability is an inherent part of the results of measurements and of the measurement process.

Measurement errors can be divided into two components: random and systematic.^[2] Random errors are errors in measurement that lead to measurable values being inconsistent when repeated measurements of a constant attribute or quantity are taken. Systematic errors are errors that are not determined by chance but are introduced by repeatable processes inherent to the system.^[3] Systematic error may also refer to an error with a non-zero mean, the effect of which is not reduced when observations are averaged.^{[citation needed]}

Measurement errors can be summarized in terms of accuracy and precision. Measurement error should not be confused with measurement uncertainty. 

https://en.wikipedia.org/wiki/Observational_error

https://en.wikipedia.org/wiki/Errors_and_residuals

https://en.wikipedia.org/wiki/Realization_(probability)

https://en.wikipedia.org/wiki/Probability

https://en.wikipedia.org/wiki/Almost_surely

 

https://en.wikipedia.org/wiki/Realization_(probability)

https://en.wikipedia.org/wiki/Probability_axioms

https://en.wikipedia.org/wiki/Probability_theory

https://en.wikipedia.org/wiki/Event_(probability_theory)

https://en.wikipedia.org/wiki/Empirical_distribution_function

https://en.wikipedia.org/wiki/Empirical_probability

https://en.wikipedia.org/wiki/Random_variable

 

https://en.wikipedia.org/wiki/Outcome_(probability)

 

https://en.wikipedia.org/wiki/Sample_space

https://en.wikipedia.org/wiki/Mutual_exclusivity

 

https://en.wikipedia.org/wiki/Collectively_exhaustive_events

 

https://en.wikipedia.org/wiki/Dice

https://en.wikipedia.org/wiki/Logic

https://en.wikipedia.org/wiki/Informal_logic

 

https://en.wikipedia.org/wiki/Measurable_space

https://en.wikipedia.org/wiki/Observational_error

 

https://en.wikipedia.org/wiki/Mistake

https://en.wikipedia.org/wiki/Statistical_parameter

 

https://en.wikipedia.org/wiki/Standard_deviation

 

https://en.wikipedia.org/wiki/Statistical_dispersion

 

https://en.wikipedia.org/wiki/Central_tendency

https://en.wikipedia.org/wiki/Arithmetic_mean

 

https://en.wikipedia.org/wiki/Observational_study

 

https://en.wikipedia.org/wiki/Randomized_controlled_trial

 

https://en.wikipedia.org/wiki/Random_assignment

https://en.wikipedia.org/wiki/Human_subject_research

https://en.wikipedia.org/wiki/Biological_specimen

https://en.wikipedia.org/wiki/Biorepository

 

https://en.wikipedia.org/wiki/Random_number_generation

 

https://en.wikipedia.org/wiki/Confounding

 

https://en.wikipedia.org/wiki/Spurious_relationship

 

https://en.wikipedia.org/wiki/Dependent_and_independent_variables

 

https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation

 

https://en.wikipedia.org/wiki/Causality

https://en.wikipedia.org/wiki/Internal_validity

 

https://en.wikipedia.org/wiki/Generalization

 

https://en.wikipedia.org/wiki/Observational_error

 

https://en.wikipedia.org/wiki/Time-invariant_system

 

https://en.wikipedia.org/wiki/Dependent_and_independent_variables

 

https://en.wiktionary.org/wiki/variation

 

https://en.wikipedia.org/wiki/Skewness

 

https://en.wikipedia.org/wiki/Unimodality

 

The standard error (SE)^[1] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution^[2] or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).^[1]

The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size.^[1] In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).

https://en.wikipedia.org/wiki/Standard_error

 

https://en.wikipedia.org/wiki/Probability_interpretations

https://en.wikipedia.org/wiki/Measurable_function

https://en.wikipedia.org/wiki/%CE%A3-algebra

https://en.wikipedia.org/w/index.php?title=Probability_measure_space&redirect=no

https://en.wikipedia.org/wiki/Conditional_probability_distribution

https://en.wikipedia.org/wiki/Sufficient_statistic

 

https://en.wikipedia.org/wiki/Kolmogorov_structure_function

 

https://en.wikipedia.org/wiki/Algorithmic_information_theory

 

https://en.wikipedia.org/wiki/Prefix_code

https://en.wikipedia.org/wiki/Variable-length_code#Uniquely_decodable_codes

 

(in the self-delimited case)

https://en.wikipedia.org/wiki/Algorithmic_information_theory

 

Unlike classical information theory, algorithmic information theory gives formal, rigorous definitions of a random string and a random infinite sequence that do not depend on physical or philosophical intuitions about nondeterminism or likelihood.

https://en.wikipedia.org/wiki/Algorithmic_information_theory

Although Ω is easily defined, in any consistent axiomatizable theory one can only compute finitely many digits of Ω, so it is in some sense unknowable, providing an absolute limit on knowledge that is reminiscent of Gödel's incompleteness theorems.  

https://en.wikipedia.org/wiki/Algorithmic_information_theory

https://en.wikipedia.org/wiki/Limit_of_a_sequence

https://en.wikipedia.org/wiki/Bayesian_inference

https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables

https://en.wikipedia.org/wiki/Rigour#Mathematical_rigour

https://en.wikipedia.org/wiki/Formal_system

https://en.wikipedia.org/wiki/Kolmogorov_complexity

https://en.wikipedia.org/wiki/Algorithmically_random_sequence

https://en.wiktionary.org/wiki/nondeterminism

https://en.wikipedia.org/wiki/Likelihood_function

https://en.wikipedia.org/wiki/Point_estimation

https://en.wikipedia.org/wiki/Mean_squared_error

https://en.wikipedia.org/wiki/Bias_of_an_estimator

https://en.wikipedia.org/w/index.php?title=Unbiased_estimator&redirect=no

https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function

 

https://en.wikipedia.org/wiki/Limit_(mathematics)

 

https://en.wikipedia.org/wiki/Net_(topology)

 

https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces

 

 https://en.wikipedia.org/wiki/Image_(mathematics)#Inverse_image

Inverse image

"Preimage" redirects here. For the cryptographic attack on hash functions, see preimage attack.

Let ${\displaystyle f}$ be a function from ${\displaystyle X}$ to ${\displaystyle Y.}$ The preimage or inverse image of a set ${\displaystyle B\subseteq Y}$ under ${\displaystyle f,}$ denoted by ${\displaystyle f^{-1}[B],}$ is the subset of ${\displaystyle X}$ defined by

$${\displaystyle f^{-1}[B]=\{x\in X:f(x)\in B\}.}$$

Other notations include ${\displaystyle f^{-1}(B)}$ and ${\displaystyle f^{-}(B).}$^[4] The inverse image of a singleton set, denoted by ${\displaystyle f^{-1}[\{y\}]}$ or by ${\displaystyle f^{-1}[y],}$ is also called the fiber or fiber over ${\displaystyle y}$ or the level set of ${\displaystyle y.}$ The set of all the fibers over the elements of ${\displaystyle Y}$ is a family of sets indexed by ${\displaystyle Y.}$

For example, for the function ${\displaystyle f(x)=x^2,}$ the inverse image of ${\displaystyle \{4\}}$ would be ${\displaystyle \{-2,2\}.}$ Again, if there is no risk of confusion, ${\displaystyle f^{-1}[B]}$ can be denoted by ${\displaystyle f^{-1}(B),}$ and ${\displaystyle f^{-1}}$ can also be thought of as a function from the power set of ${\displaystyle Y}$ to the power set of ${\displaystyle X.}$ The notation ${\displaystyle f^{-1}}$ should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of ${\displaystyle B}$ under ${\displaystyle f}$ is the image of ${\displaystyle B}$ under ${\displaystyle f^{-1}.}$

Notation for image and inverse image

The traditional notations used in the previous section do not distinguish the original function ${\displaystyle f:X\to Y}$ from the image-of-sets function ${\displaystyle f:\mathcal{P}(X)\to \mathcal{P}(Y)}$; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and usually does not cause confusion. But if needed, an alternative^[5] is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

• ${\displaystyle f^{\to }:\mathcal{P}(X)\to \mathcal{P}(Y)}$ with ${\displaystyle f^{\to }(A)=\{f(a)|a\in A\}}$
• ${\displaystyle f^{\leftarrow }:\mathcal{P}(Y)\to \mathcal{P}(X)}$ with ${\displaystyle f^{\leftarrow }(B)=\{a\in X|f(a)\in B\}}$

Star notation

• ${\displaystyle f_{\star }:\mathcal{P}(X)\to \mathcal{P}(Y)}$ instead of ${\displaystyle f^{\to }}$
• ${\displaystyle f^{\star }:\mathcal{P}(Y)\to \mathcal{P}(X)}$ instead of ${\displaystyle f^{\leftarrow }}$

Other terminology

• An alternative notation for ${\displaystyle f[A]}$ used in mathematical logic and set theory is ${\displaystyle f{}^{″}A.}$^[6][7]
• Some texts refer to the image of ${\displaystyle f}$ as the range of ${\displaystyle f,}$^[8] but this usage should be avoided because the word "range" is also commonly used to mean the codomain of ${\displaystyle f.}$

https://en.wikipedia.org/wiki/Image_(mathematics)#Inverse_image

 

Continuous functions between topological spaces

Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).

A function

$${\displaystyle f:X\to Y}$$

between two topological spaces X and Y is continuous if for every open set ${\displaystyle V\subseteq Y,}$ the inverse image

$${\displaystyle f^{-1}(V)=\{x\in X|f(x)\in V\}}$$

is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology ${\displaystyle T_X}$), but the continuity of f depends on the topologies used on X and Y.

This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.

An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions

$${\displaystyle f:X\to T}$$

to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

Continuity at a point

Continuity at a point: For every neighborhood V of ${\displaystyle f(x)}$, there is a neighborhood U of x such that ${\displaystyle f(U)\subseteq V}$

The translation in the language of neighborhoods of the ${\displaystyle (\epsilon ,\delta )}$-definition of continuity leads to the following definition of the continuity at a point:

A function ${\displaystyle f:X\to Y}$ is continuous at a point ${\displaystyle x\in X}$ if and only if for any neighborhood V of ${\displaystyle f(x)}$ in Y, there is a neighborhood U of x such that ${\displaystyle f(U)\subseteq V.}$

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, and ${\displaystyle f^{-1}(V)}$ is the largest subset U of X such that ${\displaystyle f(U)\subseteq V,}$ this definition may be simplified into:

A function ${\displaystyle f:X\to Y}$ is continuous at a point ${\displaystyle x\in X}$ if and only if ${\displaystyle f^{-1}(V)}$ is a neighborhood of x for every neighborhood V of ${\displaystyle f(x)}$ in Y.

As an open set is a set that is a neighborhood of all its points, a function ${\displaystyle f:X\to Y}$ is continuous at every point of X if and only if it is a continuous function.

If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above ${\displaystyle \epsilon -\delta }$ definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If however the target space is a Hausdorff space, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

Given ${\displaystyle x\in X,}$ a map ${\displaystyle f:X\to Y}$ is continuous at ${\displaystyle x}$ if and only if whenever ${\displaystyle \mathcal{B}}$ is a filter on ${\displaystyle X}$ that converges to ${\displaystyle x}$ in ${\displaystyle X,}$ which is expressed by writing ${\displaystyle \mathcal{B}\to x,}$ then necessarily ${\displaystyle f(\mathcal{B})\to f(x)}$ in ${\displaystyle Y.}$ If ${\displaystyle \mathcal{N}(x)}$ denotes the neighborhood filter at ${\displaystyle x}$ then ${\displaystyle f:X\to Y}$ is continuous at ${\displaystyle x}$ if and only if ${\displaystyle f(\mathcal{N}(x))\to f(x)}$ in ${\displaystyle Y.}$^[16] Moreover, this happens if and only if the prefilter ${\displaystyle f(\mathcal{N}(x))}$ is a filter base for the neighborhood filter of ${\displaystyle f(x)}$ in ${\displaystyle Y.}$^[16]

Alternative definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function ${\displaystyle f:X\to Y}$ is sequentially continuous if whenever a sequence ${\displaystyle \left(x_n\right)}$ in ${\displaystyle X}$ converges to a limit ${\displaystyle x,}$ the sequence ${\displaystyle \left(f\left(x_n\right)\right)}$ converges to ${\displaystyle f(x).}$ Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If ${\displaystyle X}$ is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ${\displaystyle X}$ is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:^[17]

Theorem — A function ${\displaystyle f:A\subseteq \mathbb{R}\to \mathbb{R}}$ is continuous at ${\displaystyle x_0}$ if and only if it is sequentially continuous at that point.

Proof

Closure operator and interior operator definitions

In terms of the interior operator, a function ${\displaystyle f:X\to Y}$ between topological spaces is continuous if and only if for every subset ${\displaystyle B\subseteq Y,}$

$${\displaystyle f^{-1}\left({\mathrm{int}}_{Y}B\right)\subseteq {\mathrm{int}}_X\left(f^{-1}(B)\right).}$$

In terms of the closure operator, ${\displaystyle f:X\to Y}$ is continuous if and only if for every subset ${\displaystyle A\subseteq X,}$

$${\displaystyle f\left({\mathrm{cl}}_{X}A\right)\subseteq {\mathrm{cl}}_Y(f(A)).}$$

That is to say, given any element ${\displaystyle x\in X}$ that belongs to the closure of a subset ${\displaystyle A\subseteq X,}$ ${\displaystyle f(x)}$ necessarily belongs to the closure of ${\displaystyle f(A)}$ in ${\displaystyle Y.}$ If we declare that a point ${\displaystyle x}$ is close to a subset ${\displaystyle A\subseteq X}$ if ${\displaystyle x\in {\mathrm{cl}}_{X}A,}$ then this terminology allows for a plain English description of continuity: ${\displaystyle f}$ is continuous if and only if for every subset ${\displaystyle A\subseteq X,}$ ${\displaystyle f}$ maps points that are close to ${\displaystyle A}$ to points that are close to ${\displaystyle f(A).}$ Similarly, ${\displaystyle f}$ is continuous at a fixed given point ${\displaystyle x\in X}$ if and only if whenever ${\displaystyle x}$ is close to a subset ${\displaystyle A\subseteq X,}$ then ${\displaystyle f(x)}$ is close to ${\displaystyle f(A).}$

Instead of specifying topological spaces by their open subsets, any topology on ${\displaystyle X}$ can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset ${\displaystyle A}$ of a topological space ${\displaystyle X}$ to its topological closure ${\displaystyle {\mathrm{cl}}_{X}A}$ satisfies the Kuratowski closure axioms. Conversely, for any closure operator ${\displaystyle A\mapsto \mathrm{cl}A}$ there exists a unique topology ${\displaystyle \tau }$ on ${\displaystyle X}$ (specifically, ${\displaystyle \tau :=\{X\setminus \mathrm{cl}A:A\subseteq X\}}$) such that for every subset ${\displaystyle A\subseteq X,}$ ${\displaystyle \mathrm{cl}A}$ is equal to the topological closure ${\displaystyle {\mathrm{cl}}_{(X,\tau )}A}$ of ${\displaystyle A}$ in ${\displaystyle (X,\tau ).}$ If the sets ${\displaystyle X}$ and ${\displaystyle Y}$ are each associated with closure operators (both denoted by ${\displaystyle \mathrm{cl}}$) then a map ${\displaystyle f:X\to Y}$ is continuous if and only if ${\displaystyle f(\mathrm{cl}A)\subseteq \mathrm{cl}(f(A))}$ for every subset ${\displaystyle A\subseteq X.}$

Similarly, the map that sends a subset ${\displaystyle A}$ of ${\displaystyle X}$ to its topological interior ${\displaystyle {\mathrm{int}}_{X}A}$ defines an interior operator. Conversely, any interior operator ${\displaystyle A\mapsto \mathrm{int}A}$ induces a unique topology ${\displaystyle \tau }$ on ${\displaystyle X}$ (specifically, ${\displaystyle \tau :=\{\mathrm{int}A:A\subseteq X\}}$) such that for every ${\displaystyle A\subseteq X,}$ ${\displaystyle \mathrm{int}A}$ is equal to the topological interior ${\displaystyle {\mathrm{int}}_{(X,\tau )}A}$ of ${\displaystyle A}$ in ${\displaystyle (X,\tau ).}$ If the sets ${\displaystyle X}$ and ${\displaystyle Y}$ are each associated with interior operators (both denoted by ${\displaystyle \mathrm{int}}$) then a map ${\displaystyle f:X\to Y}$ is continuous if and only if ${\displaystyle f^{-1}(\mathrm{int}B)\subseteq \mathrm{int}\left(f^{-1}(B)\right)}$ for every subset ${\displaystyle B\subseteq Y.}$^[18]

Filters and prefilters

Main article: Filters in topology

Continuity can also be characterized in terms of filters. A function ${\displaystyle f:X\to Y}$ is continuous if and only if whenever a filter ${\displaystyle \mathcal{B}}$ on ${\displaystyle X}$ converges in ${\displaystyle X}$ to a point ${\displaystyle x\in X,}$ then the prefilter ${\displaystyle f(\mathcal{B})}$ converges in ${\displaystyle Y}$ to ${\displaystyle f(x).}$ This characterization remains true if the word "filter" is replaced by "prefilter."^[16]

Properties

If ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to Z}$ are continuous, then so is the composition ${\displaystyle g\circ f:X\to Z.}$ If ${\displaystyle f:X\to Y}$ is continuous and

• X is compact, then f(X) is compact.
• X is connected, then f(X) is connected.
• X is path-connected, then f(X) is path-connected.
• X is Lindelöf, then f(X) is Lindelöf.
• X is separable, then f(X) is separable.

The possible topologies on a fixed set X are partially ordered: a topology ${\displaystyle {\tau }_1}$ is said to be coarser than another topology ${\displaystyle {\tau }_2}$ (notation: ${\displaystyle {\tau }_1\subseteq {\tau }_2}$) if every open subset with respect to ${\displaystyle {\tau }_1}$ is also open with respect to ${\displaystyle {\tau }_2.}$ Then, the identity map

$${\displaystyle {\mathrm{id}}_X:\left(X,{\tau }_2\right)\to \left(X,{\tau }_1\right)}$$

is continuous if and only if ${\displaystyle {\tau }_1\subseteq {\tau }_2}$ (see also comparison of topologies). More generally, a continuous function

$${\displaystyle \left(X,{\tau }_X\right)\to \left(Y,{\tau }_Y\right)}$$

stays continuous if the topology ${\displaystyle {\tau }_Y}$ is replaced by a coarser topology and/or ${\displaystyle {\tau }_X}$ is replaced by a finer topology.

Homeomorphisms

Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function ${\displaystyle f^{-1}}$ need not be continuous. A bijective continuous function with continuous inverse function is called a homeomorphism.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

Defining topologies via continuous functions

Given a function

$${\displaystyle f:X\to S,}$$

where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which ${\displaystyle f^{-1}(A)}$ is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus the final topology can be characterized as the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.

Dually, for a function f from a set S to a topological space X, the initial topology on S is defined by designating as an open set every subset A of S such that ${\displaystyle A=f^{-1}(U)}$ for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.

A topology on a set S is uniquely determined by the class of all continuous functions ${\displaystyle S\to X}$ into all topological spaces X. Dually, a similar idea can be applied to maps ${\displaystyle X\to S.}$

Related notions

If ${\displaystyle f:S\to Y}$ is a continuous function from some subset ${\displaystyle S}$ of a topological space ${\displaystyle X}$ then a continuous extension of ${\displaystyle f}$ to ${\displaystyle X}$ is any continuous function ${\displaystyle F:X\to Y}$ such that ${\displaystyle F(s)=f(s)}$ for every ${\displaystyle s\in S,}$ which is a condition that often written as ${\displaystyle f=F{|}_S.}$ In words, it is any continuous function ${\displaystyle F:X\to Y}$ that restricts to ${\displaystyle f}$ on ${\displaystyle S.}$ This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. Were ${\displaystyle f:S\to Y}$ not continuous then it could not possibly have a continuous extension. If ${\displaystyle Y}$ is a Hausdorff space and ${\displaystyle S}$ is a dense subset of ${\displaystyle X}$ then a continuous extension of ${\displaystyle f:S\to Y}$ to ${\displaystyle X,}$ if one exists, will be unique. The Blumberg theorem states that if ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ is an arbitrary function then there exists a dense subset ${\displaystyle D}$ of ${\displaystyle \mathbb{R}}$ such that the restriction ${\displaystyle f{|}_D:D\to \mathbb{R}}$ is continuous; in other words, every function ${\displaystyle \mathbb{R}\to \mathbb{R}}$ can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in order theory, an order-preserving function ${\displaystyle f:X\to Y}$ between particular types of partially ordered sets ${\displaystyle X}$ and ${\displaystyle Y}$ is continuous if for each directed subset ${\displaystyle A}$ of ${\displaystyle X,}$ we have ${\displaystyle supf(A)=f(supA).}$ Here ${\displaystyle sup}$ is the supremum with respect to the orderings in ${\displaystyle X}$ and ${\displaystyle Y,}$ respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.^[19][20]

In category theory, a functor

$${\displaystyle F:\mathcal{C}\to \mathcal{D}}$$

between two categories is called continuous if it commutes with small limits. That is to say,

$${\displaystyle \underset{i\in I}{\underleftarrow{\lim }}F(C_i)\cong F\left(\underset{i\in I}{\underleftarrow{\lim }}{C}_i\right)}$$

for any small (that is, indexed by a set ${\displaystyle I,}$ as opposed to a class) diagram of objects in ${\displaystyle \mathcal{C}}$.

A continuity space is a generalization of metric spaces and posets,^[21][22] which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.^[23]

See also

• Continuity (mathematics)
• Absolute continuity
• Dini continuity
• Equicontinuity
• Geometric continuity
• Parametric continuity
• Classification of discontinuities
• Coarse function
• Continuous function (set theory)
• Continuous stochastic process
• Normal function
• Open and closed maps
• Piecewise
• Symmetrically continuous function

• Direction-preserving function - an analogue of a continuous function in discrete spaces.

https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces

In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to analyze. It is implicit here that the index of the stochastic process is a continuous variable. Some authors^[1] define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in some terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.^[1]

^{https://en.wikipedia.org/wiki/Continuous_stochastic_process}

In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a continuous set of values, as contrasted with a discrete-time process for which the index variable takes only distinct values. An alternative terminology uses continuous parameter as being more inclusive.^[1]

A more restricted class of processes are the continuous stochastic processes; here the term often (but not always^[2]) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.^[2]

Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are called continuous-time random walks.^[3] 

https://en.wikipedia.org/wiki/Continuous-time_stochastic_process

In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.  

https://en.wikipedia.org/wiki/Discrete_time_and_continuous_time

https://en.wikipedia.org/w/index.php?title=Continuous_signal&redirect=no

https://en.wikipedia.org/wiki/Continuous_or_discrete_variable#Discrete_variable

https://en.wikipedia.org/wiki/Discrete_measure

https://en.wikipedia.org/wiki/Isolated_point

https://en.wikipedia.org/wiki/Category:General_topology

Definition

The likelihood function, parameterized by a (possibly multivariate) parameter ${\displaystyle \theta }$, is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given a probability density or mass function

${\displaystyle x\mapsto f(x\mid \theta ),}$

where ${\displaystyle x}$ is a realization of the random variable ${\displaystyle X}$, the likelihood function is

${\displaystyle \theta \mapsto f(x\mid \theta ),}$

often written

${\displaystyle \mathcal{L}(\theta \mid x).}$

In other words, when ${\displaystyle f(x\mid \theta )}$ is viewed as a function of ${\displaystyle x}$ with ${\displaystyle \theta }$ fixed, it is a probability density function, and when viewed as a function of ${\displaystyle \theta }$ with ${\displaystyle x}$ fixed, it is a likelihood function. The likelihood function does not specify the probability that ${\displaystyle \theta }$ is the truth, given the observed sample ${\displaystyle X=x}$. Such an interpretation is a common error, with potentially disastrous consequences (see prosecutor's fallacy). 

https://en.wikipedia.org/wiki/Likelihood_function

By the extreme value theorem, it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist.^[5] While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values are unknown.  

https://en.wikipedia.org/wiki/Likelihood_function

https://en.wikipedia.org/wiki/Counting_measure

https://en.wikipedia.org/wiki/Negative_definiteness

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Concave_function

https://en.wikipedia.org/wiki/Continuous_function

 

https://en.wikipedia.org/wiki/Compact_space

 

https://en.wikipedia.org/wiki/Limit_(mathematics)

https://en.wikipedia.org/wiki/Likelihood_function

https://en.wikipedia.org/wiki/Morse_theory

https://en.wikipedia.org/wiki/Nuisance_parameter

 

 

 

 

In the field of psychology, cognitive dissonance is the perception of contradictory information and the mental toll of it. Relevant items of information include a person's actions, feelings, ideas, beliefs, values, and things in the environment. Cognitive dissonance is typically experienced as psychological stress when persons participate in an action that goes against one or more of those things.^[1] According to this theory, when two actions or ideas are not psychologically consistent with each other, people do all in their power to change them until they become consistent.^[1][2] The discomfort is triggered by the person's belief clashing with new information perceived, wherein the individual tries to find a way to resolve the contradiction to reduce their discomfort.^[1][2][3]

In When Prophecy Fails: A Social and Psychological Study of a Modern Group That Predicted the Destruction of the World (1956) and A Theory of Cognitive Dissonance (1957), Leon Festinger proposed that human beings strive for internal psychological consistency to function mentally in the real world.^[1] A person who experiences internal inconsistency tends to become psychologically uncomfortable and is motivated to reduce the cognitive dissonance.^[1][2] They tend to make changes to justify the stressful behavior, either by adding new parts to the cognition causing the psychological dissonance (rationalization) or by avoiding circumstances and contradictory information likely to increase the magnitude of the cognitive dissonance (confirmation bias).^[1][2][3]

Coping with the nuances of contradictory ideas or experiences is mentally stressful. It requires energy and effort to sit with those seemingly opposite things that all seem true. Festinger argued that some people would inevitably resolve the dissonance by blindly believing whatever they wanted to believe.^[4]

^{https://en.wikipedia.org/wiki/Cognitive_dissonance}

^{https://en.wikipedia.org/wiki/Social_class}

^{https://en.wikipedia.org/wiki/Social_theory}

^{https://en.wikipedia.org/wiki/Social_phenomenon}

 

Sociological imagination is a term used in the field of sociology to describe a framework for understanding social reality that places personal experiences within a broader social and historical context.^[1]

It was coined by American sociologist C. Wright Mills in his 1959 book The Sociological Imagination to describe the type of insight offered by the discipline of sociology.^{[2]: 5, 7} Today, the term is used in introductory sociology textbooks to explain the nature of sociology and its relevance in daily life.^[1]

^{https://en.wikipedia.org/wiki/Sociological_imagination}

The imaginary (or social imaginary) is the set of values, institutions, laws, and symbols through which people imagine their social whole. It is common to the members of a particular social group and the corresponding society. The concept of the imaginary has attracted attention in anthropology, sociology, psychoanalysis, philosophy, and media studies. 

^{https://en.wikipedia.org/wiki/Imaginary_(sociology)}

 

^{https://en.wikipedia.org/wiki/Social_group}

^{https://en.wikipedia.org/wiki/Antisocial}

 

^{https://en.wikipedia.org/wiki/Isolation}

^{https://en.wikipedia.org/wiki/Solitary}

 

^{https://en.wikipedia.org/wiki/Unknown}

 

^{https://en.wikipedia.org/wiki/Uncertainty}

 

^{https://en.wikipedia.org/wiki/Stochastic}

 

^{https://en.wikipedia.org/wiki/Assimilation}

^{https://en.wikipedia.org/wiki/Bias}

^{https://en.wikipedia.org/wiki/Observational_error}

 

Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.^[1] In statistics, an error is not necessarily a "mistake". Variability is an inherent part of the results of measurements and of the measurement process.

Measurement errors can be divided into two components: random and systematic.^[2] Random errors are errors in measurement that lead to measurable values being inconsistent when repeated measurements of a constant attribute or quantity are taken. Systematic errors are errors that are not determined by chance but are introduced by repeatable processes inherent to the system.^[3] Systematic error may also refer to an error with a non-zero mean, the effect of which is not reduced when observations are averaged.^{[citation needed]}

Measurement errors can be summarized in terms of accuracy and precision. Measurement error should not be confused with measurement uncertainty. 

https://en.wikipedia.org/wiki/Observational_error

https://en.wikipedia.org/wiki/Errors_and_residuals

https://en.wikipedia.org/wiki/Realization_(probability)

https://en.wikipedia.org/wiki/Probability

https://en.wikipedia.org/wiki/Almost_surely

 

https://en.wikipedia.org/wiki/Realization_(probability)

https://en.wikipedia.org/wiki/Probability_axioms

https://en.wikipedia.org/wiki/Probability_theory

https://en.wikipedia.org/wiki/Event_(probability_theory)

https://en.wikipedia.org/wiki/Empirical_distribution_function

https://en.wikipedia.org/wiki/Empirical_probability

https://en.wikipedia.org/wiki/Random_variable

 

https://en.wikipedia.org/wiki/Outcome_(probability)

 

https://en.wikipedia.org/wiki/Sample_space

https://en.wikipedia.org/wiki/Mutual_exclusivity

 

https://en.wikipedia.org/wiki/Collectively_exhaustive_events

 

https://en.wikipedia.org/wiki/Dice

https://en.wikipedia.org/wiki/Logic

https://en.wikipedia.org/wiki/Informal_logic

 

https://en.wikipedia.org/wiki/Measurable_space

https://en.wikipedia.org/wiki/Observational_error

 

https://en.wikipedia.org/wiki/Mistake

https://en.wikipedia.org/wiki/Statistical_parameter

 

https://en.wikipedia.org/wiki/Standard_deviation

 

https://en.wikipedia.org/wiki/Statistical_dispersion

 

https://en.wikipedia.org/wiki/Central_tendency

https://en.wikipedia.org/wiki/Arithmetic_mean

 

https://en.wikipedia.org/wiki/Observational_study

 

https://en.wikipedia.org/wiki/Randomized_controlled_trial

 

https://en.wikipedia.org/wiki/Random_assignment

https://en.wikipedia.org/wiki/Human_subject_research

https://en.wikipedia.org/wiki/Biological_specimen

https://en.wikipedia.org/wiki/Biorepository

 

https://en.wikipedia.org/wiki/Random_number_generation

 

https://en.wikipedia.org/wiki/Confounding

 

https://en.wikipedia.org/wiki/Spurious_relationship

 

https://en.wikipedia.org/wiki/Dependent_and_independent_variables

 

https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation

 

https://en.wikipedia.org/wiki/Causality

https://en.wikipedia.org/wiki/Internal_validity

 

https://en.wikipedia.org/wiki/Generalization

 

https://en.wikipedia.org/wiki/Observational_error

 

https://en.wikipedia.org/wiki/Time-invariant_system

 

https://en.wikipedia.org/wiki/Dependent_and_independent_variables

 

https://en.wiktionary.org/wiki/variation

 

https://en.wikipedia.org/wiki/Skewness

 

https://en.wikipedia.org/wiki/Unimodality

 

The standard error (SE)^[1] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution^[2] or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).^[1]

The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size.^[1] In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).

https://en.wikipedia.org/wiki/Standard_error

 

https://en.wikipedia.org/wiki/Probability_interpretations

https://en.wikipedia.org/wiki/Measurable_function

https://en.wikipedia.org/wiki/%CE%A3-algebra

https://en.wikipedia.org/w/index.php?title=Probability_measure_space&redirect=no

https://en.wikipedia.org/wiki/Conditional_probability_distribution

https://en.wikipedia.org/wiki/Sufficient_statistic

 

https://en.wikipedia.org/wiki/Kolmogorov_structure_function

 

https://en.wikipedia.org/wiki/Algorithmic_information_theory

 

https://en.wikipedia.org/wiki/Prefix_code

https://en.wikipedia.org/wiki/Variable-length_code#Uniquely_decodable_codes

 

(in the self-delimited case)

https://en.wikipedia.org/wiki/Algorithmic_information_theory

 

Unlike classical information theory, algorithmic information theory gives formal, rigorous definitions of a random string and a random infinite sequence that do not depend on physical or philosophical intuitions about nondeterminism or likelihood.

https://en.wikipedia.org/wiki/Algorithmic_information_theory

Although Ω is easily defined, in any consistent axiomatizable theory one can only compute finitely many digits of Ω, so it is in some sense unknowable, providing an absolute limit on knowledge that is reminiscent of Gödel's incompleteness theorems.  

https://en.wikipedia.org/wiki/Algorithmic_information_theory

https://en.wikipedia.org/wiki/Limit_of_a_sequence

https://en.wikipedia.org/wiki/Bayesian_inference

https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables

https://en.wikipedia.org/wiki/Rigour#Mathematical_rigour

https://en.wikipedia.org/wiki/Formal_system

https://en.wikipedia.org/wiki/Kolmogorov_complexity

https://en.wikipedia.org/wiki/Algorithmically_random_sequence

https://en.wiktionary.org/wiki/nondeterminism

https://en.wikipedia.org/wiki/Likelihood_function

https://en.wikipedia.org/wiki/Point_estimation

https://en.wikipedia.org/wiki/Mean_squared_error

https://en.wikipedia.org/wiki/Bias_of_an_estimator

https://en.wikipedia.org/w/index.php?title=Unbiased_estimator&redirect=no

https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function

 

https://en.wikipedia.org/wiki/Limit_(mathematics)

 

https://en.wikipedia.org/wiki/Net_(topology)

 

https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces

 

 https://en.wikipedia.org/wiki/Image_(mathematics)#Inverse_image

Inverse image

"Preimage" redirects here. For the cryptographic attack on hash functions, see preimage attack.

Let be a function from to The preimage or inverse image of a set under denoted by is the subset of defined by

Other notations include and ^[4] The inverse image of a singleton set, denoted by or by is also called the fiber or fiber over or the level set of The set of all the fibers over the elements of is a family of sets indexed by

For example, for the function the inverse image of would be Again, if there is no risk of confusion, can be denoted by and can also be thought of as a function from the power set of to the power set of The notation should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of under is the image of under

Notation for image and inverse image

The traditional notations used in the previous section do not distinguish the original function from the image-of-sets function ; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and usually does not cause confusion. But if needed, an alternative^[5] is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

• with
• with

Star notation

• instead of
• instead of

Other terminology

• An alternative notation for used in mathematical logic and set theory is ^[6][7]
• Some texts refer to the image of as the range of ^[8] but this usage should be avoided because the word "range" is also commonly used to mean the codomain of

https://en.wikipedia.org/wiki/Image_(mathematics)#Inverse_image

 

Continuous functions between topological spaces

Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).

A function

between two topological spaces X and Y is continuous if for every open set the inverse image

is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology ), but the continuity of f depends on the topologies used on X and Y.

This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.

An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions

to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

Continuity at a point

Continuity at a point: For every neighborhood V of , there is a neighborhood U of x such that

The translation in the language of neighborhoods of the -definition of continuity leads to the following definition of the continuity at a point:

A function is continuous at a point if and only if for any neighborhood V of in Y, there is a neighborhood U of x such that

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, and is the largest subset U of X such that this definition may be simplified into:

A function is continuous at a point if and only if is a neighborhood of x for every neighborhood V of in Y.

As an open set is a set that is a neighborhood of all its points, a function is continuous at every point of X if and only if it is a continuous function.

If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If however the target space is a Hausdorff space, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

Given a map is continuous at if and only if whenever is a filter on that converges to in which is expressed by writing then necessarily in If denotes the neighborhood filter at then is continuous at if and only if in ^[16] Moreover, this happens if and only if the prefilter is a filter base for the neighborhood filter of in ^[16]

Alternative definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function is sequentially continuous if whenever a sequence in converges to a limit the sequence converges to Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:^[17]

Theorem — A function is continuous at if and only if it is sequentially continuous at that point.

Proof

Closure operator and interior operator definitions

In terms of the interior operator, a function between topological spaces is continuous if and only if for every subset

In terms of the closure operator, is continuous if and only if for every subset

That is to say, given any element that belongs to the closure of a subset necessarily belongs to the closure of in If we declare that a point is close to a subset if then this terminology allows for a plain English description of continuity: is continuous if and only if for every subset maps points that are close to to points that are close to Similarly, is continuous at a fixed given point if and only if whenever is close to a subset then is close to

Instead of specifying topological spaces by their open subsets, any topology on can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset of a topological space to its topological closure satisfies the Kuratowski closure axioms. Conversely, for any closure operator there exists a unique topology on (specifically, ) such that for every subset is equal to the topological closure of in If the sets and are each associated with closure operators (both denoted by ) then a map is continuous if and only if for every subset

Similarly, the map that sends a subset of to its topological interior defines an interior operator. Conversely, any interior operator induces a unique topology on (specifically, ) such that for every is equal to the topological interior of in If the sets and are each associated with interior operators (both denoted by ) then a map is continuous if and only if for every subset ^[18]

Filters and prefilters

Main article: Filters in topology

Continuity can also be characterized in terms of filters. A function is continuous if and only if whenever a filter on converges in to a point then the prefilter converges in to This characterization remains true if the word "filter" is replaced by "prefilter."^[16]

Properties

If and are continuous, then so is the composition If is continuous and

• X is compact, then f(X) is compact.
• X is connected, then f(X) is connected.
• X is path-connected, then f(X) is path-connected.
• X is Lindelöf, then f(X) is Lindelöf.
• X is separable, then f(X) is separable.

The possible topologies on a fixed set X are partially ordered: a topology is said to be coarser than another topology (notation: ) if every open subset with respect to is also open with respect to Then, the identity map

is continuous if and only if (see also comparison of topologies). More generally, a continuous function

stays continuous if the topology is replaced by a coarser topology and/or is replaced by a finer topology.

Homeomorphisms

Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function need not be continuous. A bijective continuous function with continuous inverse function is called a homeomorphism.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

Defining topologies via continuous functions

Given a function

where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus the final topology can be characterized as the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.

Dually, for a function f from a set S to a topological space X, the initial topology on S is defined by designating as an open set every subset A of S such that for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.

A topology on a set S is uniquely determined by the class of all continuous functions into all topological spaces X. Dually, a similar idea can be applied to maps

Related notions

If is a continuous function from some subset of a topological space then a continuous extension of to is any continuous function such that for every which is a condition that often written as In words, it is any continuous function that restricts to on This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. Were not continuous then it could not possibly have a continuous extension. If is a Hausdorff space and is a dense subset of then a continuous extension of to if one exists, will be unique. The Blumberg theorem states that if is an arbitrary function then there exists a dense subset of such that the restriction is continuous; in other words, every function can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in order theory, an order-preserving function between particular types of partially ordered sets and is continuous if for each directed subset of we have Here is the supremum with respect to the orderings in and respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.^[19][20]

In category theory, a functor

between two categories is called continuous if it commutes with small limits. That is to say,

for any small (that is, indexed by a set as opposed to a class) diagram of objects in .

A continuity space is a generalization of metric spaces and posets,^[21][22] which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.^[23]

See also

• Continuity (mathematics)
• Absolute continuity
• Dini continuity
• Equicontinuity
• Geometric continuity
• Parametric continuity
• Classification of discontinuities
• Coarse function
• Continuous function (set theory)
• Continuous stochastic process
• Normal function
• Open and closed maps
• Piecewise
• Symmetrically continuous function

• Direction-preserving function - an analogue of a continuous function in discrete spaces.

https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces

In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to analyze. It is implicit here that the index of the stochastic process is a continuous variable. Some authors^[1] define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in some terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.^[1]

^{https://en.wikipedia.org/wiki/Continuous_stochastic_process}

In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a continuous set of values, as contrasted with a discrete-time process for which the index variable takes only distinct values. An alternative terminology uses continuous parameter as being more inclusive.^[1]

A more restricted class of processes are the continuous stochastic processes; here the term often (but not always^[2]) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.^[2]

Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are called continuous-time random walks.^[3] 

https://en.wikipedia.org/wiki/Continuous-time_stochastic_process

In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.  

https://en.wikipedia.org/wiki/Discrete_time_and_continuous_time

https://en.wikipedia.org/w/index.php?title=Continuous_signal&redirect=no

https://en.wikipedia.org/wiki/Continuous_or_discrete_variable#Discrete_variable

https://en.wikipedia.org/wiki/Discrete_measure

https://en.wikipedia.org/wiki/Isolated_point

https://en.wikipedia.org/wiki/Category:General_topology

Definition

The likelihood function, parameterized by a (possibly multivariate) parameter , is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given a probability density or mass function

where is a realization of the random variable , the likelihood function is

often written

In other words, when is viewed as a function of with fixed, it is a probability density function, and when viewed as a function of with fixed, it is a likelihood function. The likelihood function does not specify the probability that is the truth, given the observed sample . Such an interpretation is a common error, with potentially disastrous consequences (see prosecutor's fallacy). 

https://en.wikipedia.org/wiki/Likelihood_function

By the extreme value theorem, it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist.^[5] While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values are unknown.  

https://en.wikipedia.org/wiki/Likelihood_function

https://en.wikipedia.org/wiki/Counting_measure

https://en.wikipedia.org/wiki/Negative_definiteness

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Concave_function

https://en.wikipedia.org/wiki/Continuous_function

 

https://en.wikipedia.org/wiki/Compact_space

 

https://en.wikipedia.org/wiki/Limit_(mathematics)

https://en.wikipedia.org/wiki/Likelihood_function

https://en.wikipedia.org/wiki/Morse_theory

https://en.wikipedia.org/wiki/Nuisance_parameter

https://en.wikipedia.org/wiki/Monotonic_function#Monotonicity_in_calculus_and_analysis

https://en.wikipedia.org/wiki/Prior_probability#Improper_priors

https://en.wikipedia.org/wiki/Simple_random_sample

https://en.wikipedia.org/wiki/Prior_probability#Uninformative_priors

https://en.wikipedia.org/wiki/Loss_function

https://en.wikipedia.org/wiki/Parameter_space

https://en.wikipedia.org/wiki/Stationary_point

https://en.wikipedia.org/wiki/Change_of_basis

https://en.wikipedia.org/wiki/Gamma_distribution

https://en.wikipedia.org/wiki/Rarefaction

In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called coordinates.

^{https://en.wikipedia.org/wiki/Change_of_basis}

 

^{https://en.wikipedia.org/wiki/Linear_map}

 ^{https://en.wikipedia.org/wiki/Multivariate_function} 

^{https://en.wikipedia.org/wiki/Matrix_inversion}

^{https://en.wikipedia.org/wiki/Square_matrix}

 

^{https://en.wikipedia.org/wiki/Change_of_basis}

^{https://en.wikipedia.org/wiki/Transpose}

^{https://en.wikipedia.org/wiki/Active_and_passive_transformation}

^{https://en.wikipedia.org/wiki/Diagonalizable_matrix}

^{https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors}

^{https://en.wikipedia.org/wiki/Orthogonal_matrix}

^{https://en.wikipedia.org/wiki/Rigid_body}