# Definition:Continuous Mapping

## 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

**$f$ is continuous at $x$** if and only if the limit $\ds \lim_{y \mathop \to x} \map f y$ exists and:

- $\ds \lim_{y \mathop \to x} \map f y = \map f x$

### 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 $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$** if and only if it is continuous at every point $x \in A_1$.

## Normed Vector Space

Let $M_1 = \struct{X_1, \norm {\,\cdot\,}_{X_1} }$ and $M_2 = \struct{X_2, \norm {\,\cdot\,}_{X_2} }$ be normed vector spaces.

Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$.

Let $a \in X_1$ be a point in $X_1$.

### Continuous at a Point

**$f$ is continuous at (the point) $a$ (with respect to the norms $\norm {\,\cdot\,}_{X_1}$ and $\norm {\,\cdot\,}_{X_2}$)** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in X_1: \norm {x - a}_{X_1} < \delta \implies \norm {\map f x - \map f a}_{X_2} < \epsilon$

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

### Continuous on a Space

$f$ is **continuous from $\struct{X_1, \norm {\,\cdot\,}_{X_1} }$ to $\struct{X_2, \norm {\,\cdot\,}_{X_2} }$** if and only if it is continuous at every point $x \in X_1$.

## Topology

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

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ 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 open set $U_2$ of $T_2$ such that $\map f x \in U_2$, there exists an open set $U_1$ of $T_1$ such that $x \in U_1$ and $f \sqbrk {U_1} \subseteq U_2$.

### Definition using Neighborhoods

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 $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.

### Definition using Filters

The mapping $f$ is **continuous at (the point) $x$** if and only if:

- for any filter $\FF$ on $T_1$ that converges to $x$, the corresponding image filter $f \sqbrk \FF$ converges to $\map f x$.

If necessary, we can say that **$f$ is $\tuple {\tau_1, \tau_2}$-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:

- $U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$

where $f^{-1} \sqbrk U$ denotes the preimage of $U$ under $f$.

## Sources

- 1919: Horace Lamb:
*An Elementary Course of Infinitesimal Calculus*(3rd ed.) ... (next): Chapter $\text I$. Continuity: $1$. Continuous Variation