TITLE: Differentiablility over closed intervals
QUESTION [31 upvotes]: In calculus of one variable I read:
A function $f$ is differentiable on an open interval if it is differentiable at every number of the interval.
I wonder why in the definition we suppose that the interval is open. This is the case in Rolle theorem, Mean value theorem,etc.

Do we have a notion of a function being differentiable on a closed interval $[a,b]$?

REPLY [3 votes]: Nice question, Palio, I was searching the net for the same question and it's answer, here is what I understand, take example of derivative, derivative by definition is $\frac{f(x+h)-f(x)}{h}$, here $h$ can be positive or negative, for the term $f(x+h)$ to make sense or be valid, the function $f$ must be defined "around" the point $x$, this mathematically is defining a function on an open interval.
Hope this helps, although 4 months have passed since you asked :)