Definition:Continuous Real Function

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.

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.

It is worth addressing each type of interval in turn.

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.