Suppose $f:[a,b]\rightarrow \mathbb R $ is differentiable on (a,b) and continuous on [a,b]. Does it follow that $f$ is right-differentiable at $a$ and left-differentiable at $b$?
I guess it does not follow since $f:[a,b]\rightarrow \mathbb R $ can be differentiable at a point, call it $ c \in (a,b)$ but not at the end points since the interval is open.
Is it right? If so, how do i formally prove?