Definition:Continuous Real Function
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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.
Formal Definition
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.
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:
- continuous at every point of $\left({a .. b}\right)$;
- continuous on the left at $b$;
- 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 as well. However, because we only have "access" to the function on one side of each end point, all we can do is insist on continuity on the side of the end point that 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:
- continuous at every point of $\left({a .. b}\right)$;
- 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:
- continuous at every point of $\left({a .. b}\right)$;
- 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.
Sources
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{III}$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definitions $1.4.3, \ 1.4.4$
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.6$
