Assume $f:[a,b] \to \mathbb R$ is convex. We want to show $f(a+)=\displaystyle \lim_{x \to a^+} f(x)$ and $f(b-)=\displaystyle \lim_{x \to b^-} f(x)$ exist. I was thinking of first showing that $f$ has right and left derivatives everywhere in $(a,b)$ which are increasing, then use that is $x
It seems that this is a bit too complicated though... Is there a simpler way to do it?