Can a function be differentiable at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a,b]$. Is it possible that $f$ is differentiable on the closed interval $[a,b]$, or must the maximal domain for $f'$ be $(a,b)$?

Solution 1:

It depends on the definition of differentiability you have. Some textbooks only define it for interior points. But there is also a more general definition (see this answer for references):

A function $f: A \rightarrow \mathbb R$ is differentiable for an accumulation point $a \in A$ of $A$ with derivative $f^\prime(a)$, iff for each $x_n \rightarrow a$ with $x_n\in A\setminus\{a\}$ you have $f^\prime(a) = \lim_{n\rightarrow\infty} \frac{f(x_n)-f(a)}{x_n-a}$.

So the answer is yes: You can define the derivative in a way, such that $f'$ is also defined for the end points of a closed interval. Note that for some theorem like the mean value theorem you only need continuity at the end points of the interval.