Definition:Continuous Mapping

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Definition

The concept of continuity makes precise the intuitive notion that a function has no "jumps" at a given point.

Loosely speaking, in the case of a real function, continuity at a point is defined as the property that the graph of the function does not have a "break" at the point.

This concept appears throughout mathematics and correspondingly has many variations and generalizations.


Real Function

Continuity at a Point

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.

Let $x \in A$ be a point of $A$.


Then $f$ is continuous at $x$ if and only if the limit $\displaystyle \lim_{y \to x} f \left({y}\right)$ exists and:

$\displaystyle \lim_{y \to x} \ f \left({y}\right) = f \left({x}\right)$


Continuous Everywhere

Let $f : \R \to \R$ be a real function.


Then $f$ is everywhere continuous if and only if $f$ is continuous at every point in $\R$.


Continuity on a Subset of Domain

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.


Then $f$ is continuous on $A$ if and only if $f$ is continuous at every point of $A$.


Complex Function

As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces.


Metric Space

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

Let $a \in A_1$ be a point in $A_1$.


Continuous at a Point

$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$) if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$

where $\R_{>0}$ denotes the set of all strictly positive real numbers.


Continuous on a Space

$f$ is continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$ if and only if it is continuous at every point $x \in A_1$.


Topology

The most general definition of continuity is the concept as defined in a topological space.


Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.


Continuous at a Point

Let $x \in S_1$.

Definition using Open Sets

The mapping $f$ is continuous at (the point) $x$ (with respect to the topologies $\tau_1$ and $\tau_2$) if and only if:

For every neighborhood $N$ of $f \left({x}\right)$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \left({M}\right) \subseteq N$.


Definition using Filters

The mapping $f$ is continuous at (the point) $x$ if and only if for any filter $\mathcal F$ on $T_1$ that converges to $x$, the corresponding image filter $f \left({\mathcal F}\right)$ converges to $f \left({x}\right)$.


If necessary, we can say that $f$ is $\left({\tau_1, \tau_2}\right)$-continuous at $x$.


Continuous on a Set

Let $S$ be a subset of $S_1$.

The mapping $f$ is continuous on $S$ if and only if $f$ is continuous at every point $x \in S$.


Continuous Everywhere

Definition by Pointwise Continuity

The mapping $f$ is continuous everywhere (or simply continuous) if and only if $f$ is continuous at every point $x \in S_1$.


Definition by Open Sets

The mapping $f$ is continuous on $S_1$ if and only if:

$U \in \tau_2 \implies f^{-1} \left({U}\right) \in \tau_1$

where $f^{-1} \left({U}\right)$ denotes the preimage of $U$ under $f$.