Definition:Continuous Real Function/Closed Interval

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Let $f$ be a real function defined on a closed interval $\closedint a b$.

$f$ is continuous on $\closedint a b$ if and only if it is:

$(1): \quad$ continuous at every point of the open interval $\openint a b$
$(2): \quad$ 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.


Example: $\dfrac 1 {1 + e^{1 / x} }$ on $\closedint 0 1$

Consider the real function $f$ defined as:

$f := \begin {cases} \dfrac 1 {1 + e^{1 / x} } & : x \ne 0 \\ 0 & : x = 0 \end {cases}$

Then $f$ is continuous on the closed interval $\closedint 0 1$.

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