# Definition:Continuous Real Function/Closed Interval/Definition 1

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## Definition

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

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.

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.2$: Examples - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 9.1$