I am reading about differentiability of functions of the form $f$ : $[a,b] \to \mathbb{R}$ where f is continuous on the interval. In many of the theorems we usually say that there exists a point $c$ $\in$ $(a,b)$ such that some property of $f$ or $f’$ holds at $c$. Why is it that we usually look at the points inside the interval and don’t say anything about the end points ? An example of this is the mean value theorem. Thanks!
Edit: I know this is quite a general question I was hoping for some intuition on such problems.
The reason is that two-sided limits don't technically exist at the endpoints. You need limits to talk about derivatives, since they're defined in terms of limits. Now some authors define 'differentiable on a closed interval $[a,b]$' as differentiable on the open interval $(a,b)$ as well as differentiable from the right at $a,$ and differentiable from the left at $b$. You can look those definitions up on the wiki. The same goes for continuity.