# Definition:Continuity/Real Function

## Contents

## Definition

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

Then **$f$ is continuous on $\R$** iff $f$ is continuous at every point of $\R$.

### Continuity at a Point

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

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

Then **$f$ 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 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 $A$** iff $f$ is continuous at every point of $A$.

## Continuity from One Side

### Continuity from the Left at a Point

Let $x_0 \in A$.

Then $f$ is said to be **left-continuous at $x_0$** iff the limit from the left of $f \left({x}\right)$ as $x \to x_0$ exists and:

- $\displaystyle \lim_{\substack{x \mathop \to x_0^- \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$

where $\displaystyle \lim_{x \mathop \to x_0^-}$ is a limit from the left.

### Continuity from the Right at a Point

Let $x_0 \in S$.

Then $f$ is said to be **right-continuous at $x_0$** iff the limit from the right of $f \left({x}\right)$ as $x \to x_0$ exists and:

- $\displaystyle \lim_{\substack{x \mathop \to x_0^+ \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$

where $\displaystyle \lim_{x \mathop \to x_0^+}$ is a limit from the right.

### Continuity on an Interval

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

It is worth addressing each type of interval in turn.

#### Open Interval

This is a straightforward application of continuity on a set.

Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.

Then $f$ is **continuous on $\left({a \,.\,.\, b}\right)$** iff it is continuous at every point of $\left({a \,.\,.\, b}\right)$.

#### Closed Interval

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Then $f$ is **continuous on $\left[{a \,.\,.\, b}\right]$** iff it is:

- $(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
- $(2): \quad$ continuous on the left at $b$
- $(3): \quad$ continuous on the right at $a$.

That is, if $f$ is to be continuous over the *whole* of a closed interval, it needs to be continuous at the end points.

Because we only have "access" to the function on one side of each end points, all we can do is insist on continuity on the side of the end points on which the function is defined.

#### Half Open Intervals

Similar definitions apply to half open intervals:

Let $f$ be a real function defined on a half open interval $\left({a \,.\,.\, b}\right]$.

Then $f$ is **continuous on $\left({a \,.\,.\, b}\right]$** iff it is:

- $(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
- $(2): \quad$ continuous on the left at $b$.

Let $f$ be a real function defined on a half open interval $\left[{a \,.\,.\, b}\right)$.

Then $f$ is **continuous on $\left[{a \,.\,.\, b}\right)$** iff it is:

- $(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
- $(2): \quad$ continuous on the right at $a$.

## As a Metric Space

Note that the definition for continuity at a point as given here is the same as that for a metric space, where the usual metric is taken on the real number line.

## Informal Definition

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

Loosely speaking, a real function is **continuous** at a point if the graph of the function does not have a "break" at the point.

## Sources

- George McCarty:
*Topology: An Introduction with Application to Topological Groups*(1967)... (previous)... (next): $\text{III}$: The Definition - W.A. Sutherland:
*Introduction to Metric and Topological Spaces*(1975)... (previous)... (next): $\S 1.4$: Continuity: Definition $1.4.4$