Definition:Continuous Real Function/Interval

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. However, 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:
 * 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 $a$.

Warning: Domain of Function

 * The limit in the previous definitions must be taken among points inside the domain $A$ of the function $f$.

For example, if $A$ is a closed interval $\left[{a \,.\,.\, b}\right]$, then to say that $f$ is continuous at $a$ means that
 * $\displaystyle \lim_{y \to a^+} f \left({y}\right) = f \left({a}\right)$

The limit must be taken from the right, as $f$ is not defined on the left of $a$.