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\begin{document}

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\section*{continuous map}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{continuous_maps}{}\section*{{Continuous maps}}\label{continuous_maps}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{EpsilonticDefinition}{The epsilontic definition for metric spaces}\dotfill \pageref*{EpsilonticDefinition} \linebreak
\noindent\hyperlink{for_topological_spaces}{For topological spaces}\dotfill \pageref*{for_topological_spaces} \linebreak
\noindent\hyperlink{further_variants}{Further variants}\dotfill \pageref*{further_variants} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{properties_preserved}{Properties preserved}\dotfill \pageref*{properties_preserved} \linebreak
\noindent\hyperlink{special_maps}{Special maps}\dotfill \pageref*{special_maps} \linebreak
\noindent\hyperlink{special_cases_in_specific_contexts}{Special cases in specific contexts}\dotfill \pageref*{special_cases_in_specific_contexts} \linebreak
\noindent\hyperlink{in_constructive_mathematics}{In constructive mathematics}\dotfill \pageref*{in_constructive_mathematics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A [[function]] $f \colon X \to Y$ is called \emph{continuous} if its values $f(x)$ do not ``jump'' with variation of its argument $x$, unless $x$ itself ``jumps''. Roughly speaking, if $x_1 \approx x_2$, then $f(x_1) \approx f(x_2)$. (This can be made into a precise definition in [[nonstandard analysis]] if care is taken about the domains of these variables.)

In order to make this precise (in standard analysis) one needs some concept of [[neighbourhoods]] of elements of $X$ and $Y$.

For instance if $X$ and $Y$ carry structure of [[metric spaces]], then one may say that $f$ is continuous if for every point $x \in X$ and for every small [[open ball]] around its image $f(x)$ in $Y$, there exists a sufficiently small open ball around $x \in X$ which is still mapped by $f$ into that target open ball. This definition turns out to have more elegant formulation that needs to mention neither the points of $x$ nor the radii of open balls around points: the [[metric]] induces a concept of [[open subsets]] and $f$ is continuous precisely if [[preimages]] under $f$ of [[open subsets]] in $Y$ are still open subsets in $X$.

This then is the general definition of continuity of a function $f$ between [[topological spaces]]:

A function between [[topological spaces]] is \emph{continuous} precisely if its [[preimages]] of [[open subsets]] are again open subsets.

Continuous maps are the [[homomorphisms]] between [[topological spaces]]. In other words, the collection of [[topological spaces]] forms a [[category]], often denoted \emph{[[Top]]}, whose [[morphisms]] are the continuous functions.

Further generalization of the concept of continuity exists, for instance to \emph{[[locales]]} (and then to [[toposes]]) or to \emph{[[convergence spaces]]}. (See also at \emph{[[continuous space]]}.)

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{EpsilonticDefinition}{}\subsubsection*{{The epsilontic definition for metric spaces}}\label{EpsilonticDefinition}

We state the definition of continuity in terms of [[epsilontic analysis]], definition \ref{EpsilonDeltaDefinitionOfContinuity} below. First recall the relevant concepts:

\begin{defn}
\label{MetricSpace}\hypertarget{MetricSpace}{}
A \emph{[[metric space]]} is

\begin{enumerate}%
\item a [[set]] $X$ (the ``underlying set'');


\item a [[function]] $d \;\colon\; X \times X \to [0,\infty)$ (the ``distance function'') from the [[Cartesian product]] of the set with itself to the [[nonnegative number|non-negative]] [[real numbers]]



\end{enumerate}
such that for all $x,y,z \in X$:

\begin{enumerate}%
\item $d(x,y) = 0 \;\Leftrightarrow\; x = y$


\item (symmetry) $d(x,y) = d(y,x)$


\item ([[triangle inequality]]) $d(x,y)+ d(y,z) \geq d(x,z)$.



\end{enumerate}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Every [[normed vector space]] $(V, {\Vert -\Vert})$ becomes a [[metric space]] according to def. \ref{MetricSpace} by setting

\begin{displaymath}
d(x,y) \coloneqq {\Vert x-y\Vert}
  \,.
\end{displaymath}
\end{example}
\begin{defn}
\label{OpenBalls}\hypertarget{OpenBalls}{}
Let $(X,d)$, be a [[metric space]]. Then for every element $x \in X$ and every $\epsilon \in \mathbb{R}_+$ a [[positive number|positive]] [[real number]], write

\begin{displaymath}
B^\circ_x(\epsilon) 
    \;\coloneqq\;
  \left\{
    y \in X \;\vert\; d(x,y) \lt \epsilon
  \right\}
\end{displaymath}
for the [[open ball]] of [[radius]] $\epsilon$ around $x$.

\end{defn}
\begin{defn}
\label{EpsilonDeltaDefinitionOfContinuity}\hypertarget{EpsilonDeltaDefinitionOfContinuity}{}
\textbf{(epsilontic definition of continuity)}

For $(X,d_X)$ and $(Y,d_Y)$ two [[metric spaces]] (def. \ref{MetricSpace}), then a [[function]]

\begin{displaymath}
f \;\colon\; X \longrightarrow Y
\end{displaymath}
is said to be \emph{continuous at a point $x \in X$} if for every $\epsilon \gt 0$ there exists $\delta\gt 0$ such that

\begin{displaymath}
d_X(x,y) \lt \delta \;\Rightarrow\; d_Y(f(x), f(y)) \lt \epsilon
\end{displaymath}
or equivalently such that

\begin{displaymath}
f(B_x^\circ(\delta)) \subset B^\circ_{f(x)}(\epsilon)
\end{displaymath}
where $B^\circ$ denotes the [[open ball]] (definition \ref{OpenBalls}).

The function $f$ is called just \emph{continuous} if it is continuous at every point $x \in X$.

\end{defn}
This definition is equivalent to a more abstract one, which does not explicitly refer to points or radii anymore:

\begin{defn}
\label{OpenSubsetsOfAMetricSpace}\hypertarget{OpenSubsetsOfAMetricSpace}{}
Let $(X,d)$ be a [[metric space]] (def. \ref{MetricSpace}). Say that

\begin{enumerate}%
\item A \emph{[[neighbourhood]]} of a point $x \in X$ is a [[subset]] $x \in U \subset X$ which contains some [[open ball]] $B_x^\circ(\epsilon)$ around $x$ (def. \ref{OpenBalls}).


\item An \emph{[[open subset]]} of $X$ is a [[subset]] $U \subset X$ such that for every for $x \in U$ it also contains a [[neighbourhood]] of $x$.



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The collection of open subsets in def. \ref{OpenSubsetsOfAMetricSpace} constitutes a \emph{[[topological space|topology]]} on the set $X$, making it a \emph{[[topological space]]}. This is called the \emph{[[metric topology]]}. Stated more concisely: the [[open balls]] in a metric space constitute the [[basis of a topology]] for the [[metric topology]].

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
A [[function]] $f \colon X \to Y$ between [[metric spaces]] (def. \ref{MetricSpace}) is continuous in the [[epsilontic analysis|epsilontic]] sense of def. \ref{EpsilonDeltaDefinitionOfContinuity} precisely if it has the property that its [[pre-images]] of [[open subsets]] of $Y$ (in the sense of def. \ref{OpenSubsetsOfAMetricSpace}) are open subsets of $X$.

\end{prop}
\begin{proof}
First assume that $f$ is continuous in the epsilontic sense. Then for $O_Y \subset Y$ any [[open subset]] and $x \in f^{-1(O_Y)}$ any point in the pre-image, we need to show that there exists a [[neighbourhood]] of $x$ in $U$. But by assumption there exists an [[open ball]] $B_x^\circ(\epsilon)$ with $f(B_X^\circ(\epsilon)) \subset O_Y$. Since this is true for all $x$, by definition this means that $f^{-1}(O_Y)$ is open in $X$.

Conversely, assume that $f^{-1}$ takes open subsets to open subsets. Then for every $x \in X$ and $B_{f(x)}^\circ(\epsilon)$ an [[open ball]] around its image, we need to produce an open ball $B_x^\circ(\delta)$ in its pre-image. But by assumption $f^{-1}(B_{f(x)}^\circ(\epsilon))$ contains a [[neighbourhood]] of $x$ which by definition means that it contains such an open ball around $x$.

\end{proof}
\hypertarget{for_topological_spaces}{}\subsubsection*{{For topological spaces}}\label{for_topological_spaces}

\begin{defn}
\label{topological}\hypertarget{topological}{}
A [[function]] $f \;\colon\; X\to Y$ between [[topological spaces]] is a \textbf{continuous map} (or is said to be \emph{continuous}) if for every [[open subset]] $U \subset Y$, the [[preimage]] $f^{-1}(U)$ is an open subset $X$.

\end{defn}
In [[nonstandard analysis]], this is equivalent to

\begin{defn}
\label{nonstandard}\hypertarget{nonstandard}{}
A [[function]] $f \;\colon\; X\to Y$ between [[topological spaces]] is a \textbf{continuous map} (or is said to be \emph{continuous}) if for every [[standard point]] $x_1$ and every [[hyperpoint]] $x_2$, if $x_1$ and $x_2$ are [[adequality|adequal]] (infinitely close, or in other words if $x_2$ is in the [[halo]] of $x_1$), then $f(x_1)$ and $\multiscripts{^*}f{}(x_2)$ are adequal (where $\multiscripts{^*}f{}$ is the [[nonstandard extension]] of $f$). Equivalently, $f$ is continuous iff $\multiscripts{^*}f{}$ is [[microcontinuous function|microcontinuous]].

\end{defn}
\hypertarget{further_variants}{}\subsubsection*{{Further variants}}\label{further_variants}

A function $f$ between [[convergence spaces]] is \textbf{continuous} if for any [[filter]] $F$ such that $F \to x$, it follows that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A) \;|\; A \in F\}$.

A \textbf{continuous map} between [[locales]] is simply a [[frame]] [[homomorphism]] in the opposite direction. Equivalently (via the [[adjoint functor theorem]]), it may be defined as a homomorphism of [[inflattices]] whose [[left adjoint]] preserves finitary [[meets]] (and hence is a frame homomorphism).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Since continuity is defined in terms of \emph{preservation of property} (namely preserving ``openness'' under preimages), it is natural to ask what other properties they preserve.\newline Also, when a property is not always preserved it is useful to label those maps which do preserve it for closer study.

\hypertarget{properties_preserved}{}\subsubsection*{{Properties preserved}}\label{properties_preserved}

\begin{enumerate}%
\item By definition, the preimage of an open set is open.


\item Similarly, the preimage of an [[closed set]] is closed.


\item The [[image]] of a [[connected space|connected subset]] is again connected.


\item The image of a [[compact space|compact subset]] is again compact (see at \emph{[[continuous images of compact subsets are compact]]})



\end{enumerate}
\hypertarget{special_maps}{}\subsubsection*{{Special maps}}\label{special_maps}

\begin{enumerate}%
\item The preimage of a compact set need not be compact; a continuous map for which this is true is known as a \textbf{[[proper map]]}.


\item The image of an open set need not be open; a continuous map for which this is true is said to be an \textbf{[[open map]]}. (Technically, an open map is any [[function]] with just this property.)


\item The image of an closed set need not be closed; a continuous map for which this is true is said to be an \textbf{[[closed map]]}. (Technically, a closed map is any function with just this property.)


\item A continuous map of topological spaces which is invertible as a function of sets is a \textbf{[[homeomorphism]]} if the [[inverse function]] is a continuous map as well.



\end{enumerate}
\hypertarget{special_cases_in_specific_contexts}{}\subsubsection*{{Special cases in specific contexts}}\label{special_cases_in_specific_contexts}

Although these don't make sense for arbitrary topological spaces (convergence spaces, locales, etc), they are special kinds of continuous maps in contexts such as [[metric spaces]]:

\begin{itemize}%
\item [[uniformly continuous maps]],
\item [[Lipschitz maps]],
\item [[short maps]],
\item [[differentiable maps]],
\item [[smooth maps]].

\end{itemize}
\hypertarget{in_constructive_mathematics}{}\subsection*{{In constructive mathematics}}\label{in_constructive_mathematics}

Various notions of continuous function are used in [[constructive mathematics]]. A function $f$ (say [[real number|real]]-valued and defined on a real [[interval]]) is:

\begin{itemize}%
\item \emph{pointwise-continuous} if it continuous in the usual [[epsilon-delta]] (or equivalently [[open-subset]]) sense;
\item \emph{uniformly continuous} if it [[uniformly continuous map|uniformly continuous]] in the usual epsilon-delta (or equivalently [[entourage]]-theoretic) sense;
\item \emph{Bishop-continuous} if it is pointwise continuous and furthermore, the restriction to any closed and bounded interval is uniformly continuous;
\item \emph{Bridges-continuous} if \ldots{} (this one's kind of complicated).

\end{itemize}
In [[classical mathematics]], these are all equivalent when the domain is itself a closed and bounded interval, and all of them except for uniform continuity are equivalent in general. The same equivalences hold in [[intuitionistic mathematics]], thanks to the [[fan theorem]]. But no two of these are equivalent in [[Russian constructivism]].

In fact, assuming that $\mathbb{R}$ is defined as the set of located [[Dedekind cuts]], there is the following negative result by [[Frank Waaldijk]] (\hyperlink{Waaldijk2003}{Waaldijk2003}): Without the [[fan theorem]], there is no notion of continuity for set-theoretic functions in [[constructive mathematics]], spelled ``kontinuity'' in the following, such that all of the following desiderata are met:

\begin{itemize}%
\item A function $[0,1] \to \mathbb{R}$ is kontinuous if and only if it is [[uniformly continuous]] in the usual sense.
\item The composition of kontinuous functions is kontinuous.
\item The function $\mathbb{R}^+ \to \mathbb{R}, x \mapsto 1/x$ is kontinuous.

\end{itemize}
The key problem is that a uniformly continuous, [[positive number|positive]]-valued function defined on $[0,1]$ might fail to be bounded below by a positive number, since the interval $[0,1]$ might fail to be [[compact space|compact]], yet its reciprocal (if also uniformly continuous) must be bounded above.

Waaldijk's negative result can be circumvented by dropping the insistence on points and instead working with maps between [[locales]], [[toposes]], or formal spaces as studied in [[formal topology]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[degree of a continuous function]]


\item [[equicontinuous family of functions]]


\item [[analytic function]]


\item [[differentiable function]]


\item [[smooth function]]


\item [[convex function]]


\item [[integrable function]], [[square-integrable function]]


\item [[bounded function]]


\item [[compactly supported function]]


\item [[measurable function]]


\item [[rapidly decreasing function]]


\item [[function with rapidly decreasing partial derivatives]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Frank Waaldijk]], \emph{On the foundations of constructive mathematics -- especially in relation to the theory of continuous functions}, 2003 (\href{http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf}{pdf})

\end{itemize}
[[!redirects continuous map]] [[!redirects continuous maps]] [[!redirects continuous function]] [[!redirects continuous functions]]

[[!redirects continuity]]



\end{document}