Definition:Continuous Real Function/Interval

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Definition

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 $\left({a \,.\,.\, b}\right)$ if and only if it is continuous at every point of $\left({a \,.\,.\, b}\right)$.

Closed Interval

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

Definition 1

The function $f$ is continuous on $\left[{a \,.\,.\, b}\right]$ if and only if it is:

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

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 point, all we can do is insist on continuity on the side of the end points on which the function is defined.

Definition 2

The function $f$ is continuous on $\left[{a \,.\,.\, b}\right]$ if and only if it is continuous at every point of $\left[{a \,.\,.\, b}\right]$.

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 $\left({a \,.\,.\, b}\right]$ if and only if 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 $\left[{a \,.\,.\, b}\right)$ if and only if it is:

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