Definition:Continuity

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.

Continuous Real Function

Continuity at a Point

Let $A \subseteq \R$ be any subset of the real numbers, and $f: A \to \R$ be a function.

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

We say that $f$ is continuous at $x$ when the limit of $f \left({y}\right)$ as $y \to x$ exists and:

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

Continuity on a Set

Let $A \subseteq \R$ be any subset of the real numbers, and $f: A \to \R$ be a function.

We say that $f$ is continuous on $A$ if $f$ is continuous at every point of $A$.

Continuity on a Singleton

• The set $A$ can be any set, but there is a case in which the definition is trivial:

if $x \in A$ is an isolated point of $A$, then every function $f: A \to \R$ is continuous at $x$, as the limit in this case is trivially equal to $f \left({x}\right)$.

Continuity from One Side

There is a related concept of continuity where one only approaches the point $x$ only from the right or from the left:

Continuity from the Left at a Point

We say that $f$ is continuous from the left at $x$ when the limit from the left of $f \left({y}\right)$ as $y \to x$ exists and:

$\displaystyle \lim_{\underset{y \in A}{y \to x^-}} f \left({y}\right) = f \left({x}\right)$

Continuity from the Right at a Point

We say that $f$ is continuous from the right at $x$ when the limit from the right of $f \left({y}\right)$ as $y \to x$ exists and:

$\displaystyle \lim_{\underset{y \in A}{y \to x^+}} f \left({y}\right) = f \left({x}\right)$

Continuity on an Interval

Where $A$ is a real interval, it is considered as a specific example of continuity on a set.

Continuous on a 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

Definition using Limit

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

• The limit of $f \left({x}\right)$ as $x \to a$ exists
• $\displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$.

Epsilon-Delta Definition

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

$\forall \epsilon > 0: \exists \delta > 0: d_1 \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$

Epsilon-Neighborhood Definition

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

$\forall N_\epsilon \left({f \left({a}\right)}\right): \exists N_\delta \left({a}\right): f \left({ N_\delta \left({a}\right)}\right) \subseteq N_\epsilon \left({f \left({a}\right)}\right)$

where $N_\epsilon \left({a}\right)$ is the $\epsilon$-neighborhood of $a$ in $M_1$.

That is, for every $\epsilon$-neighborhood of $f \left({a}\right)$ in $M_2$, there exists a $\delta$-neighborhood of $a$ in $M_1$ whose image is a subset of that $\epsilon$-neighborhood.

Continuous on a Space

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

Open Set Definition

$f$ is continuous iff:

for every set $U \subseteq M_2$ which is open in $M_2$, $f^{-1} \left({U}\right)$ is open in $M_1$.

Continuous on a Metric Subspace

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 $Y \subseteq A_1$.

By definition, $\left({Y, d_Y}\right)$ is a metric subspace of $A_1$.

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

Then $f$ is $\left({d_Y, d_2}\right)$-continuous iff:

$\forall \epsilon > 0: \exists \delta > 0: d_Y \left({x, a_1}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a_1}\right)}\right) < \epsilon$

Continuous Complex Function

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

Continuous Mapping

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

Let $T_1 = \left({A_1, \vartheta_1}\right)$ and $T_2 = \left({A_2, \vartheta_2}\right)$ be topological spaces.

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

Then $f$ is continuous (with respect to the topologies $\vartheta_1$ and $\vartheta_2$) iff:

$U \in \vartheta_2 \implies f^{-1} \left({U}\right) \in \vartheta_1$

If necessary, we can say that $f$ is $\left({\vartheta_1, \vartheta_2}\right)$-continuous.

Continuous at a Point

Let $T_1 = \left({A_1, \vartheta_1}\right)$ and $T_2 = \left({A_2, \vartheta_2}\right)$ be topological spaces.

Let $x \in T_1$.

Let $N \subseteq T_2$ be a neighborhood of $f \left({x}\right)$.

Then $f$ is continuous at (the point) $x$ iff there always exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$.

The general definition for continuous mapping follows from the definition of continuity at a point for all points in the topology.