Derivative at end point of a interval

Question

While studying Rolle's Theorem, a question came in my mind that can there exist a function which is continuous in the interval [a,b] and differentiable in the interval (a,b) but not differentiable at either one of the end points of the interval [a,b]?

Answer 1

of course let me show you an example. $$f: [1,2] \longrightarrow \mathbb{R}. $$

$$f(x) = \sqrt{(1-x)(x-2)}.$$

This function is clearly continuos in $[1,2]$ and differentiable in $(1,2)$ but using the limit definition of differentiation in $1,2$ the limit does not converges.