Let $f:(a,b) \to \mathbb{R}$ be a convex function. A convex function is absolutely continuous on each closed subinterval of $(a,b)$. Also, the right and left-hand derivatives exist at each point of $(a,b)$, and are equal to each other except on a countable set.
My question is, if a convex function is defined as $f:[a,b] \to \mathbb{R}$, is the function now absolutely continuous on $[a,b]$ ? Also, what can we say about the right and left hand derivatives on $a$ and $b$?
A convex function may be discontinuous at the endpoints of its domain.
Consider $$f(x)=\begin{cases} 1 & x = a \\ 0 & a < x < b \\ 1 & x = b \end{cases}$$