Math CentralQuandaries & Queries


Question from Pragya, a student:

The trouble I'm having is as follows :
a continuous function is most of the times defined on a closed interval,
but how is it possible to define it on a closed interval ,because to be continuous at endpoints of the interval the function's
limit must exist at that endpoint,for which it has to be defined in its neighborhood,but we don't know anything about whether the function is always defined in the neighborhood.
Please help...

HI Pragya,

If $a$ and $b$ are real numbers with $a < b$ then a function $f$ defined on $[a, b]$ is continuous if

\[\lim_{x \rightarrow c} f(x) = f(c)\]

for each $c$ with $a < c < b,$

\[\lim_{x \rightarrow a^+} f(x) = f(a),\]


\[\lim_{x \rightarrow b^-} f(x) = f(b).\]

If the last two limits don't make sense to you look up one sided limits in your textbook.

Write back if you need more assistance,

Pragya wrote back

I am confused because a continuous function is always defined on a closed interval,with the continuity at end-points being checked through one-sided limits.
but then why is differentiability defined on an open interval ,knowing it can be defined on a closed interval in similar manner as we do in continuity....


You are correct that differentiability can be defined on a closed interval by using one sided limits. Some calculus text books do introduce one sided derivatives but many do not. Most of the applications of differentiation apply to two sided derivatives and as a result authors don't take the additional effort to introduce one sided derivatives.

So why go to the extra effort to define continuity from the left and from the right in order to define continuity on a closed interval? The reason is that many of the important theorems in calculus are theorems about continuous functions on a closed interval. Three of these theorems are The Extreme Value Theorem, The intermediate Value Theorem and The Mean Value theorem.


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