TITLE: Convex function on closed bounded Interval Implies Lipschitz counterexample
QUESTION [0 upvotes]: I am considering a convex function $f:[a,b] \rightarrow \mathbb{R}$ and have been asked to show that $f$ is absolutely continuous on $[a,b]$. I've attempted to use the Chordal Slope Theorem to show that $f$ is Lipschitz on $[a,b]$, but have only been able to show that $f$ is Lipschitz for $[c,d] \subseteq (a,b)$. I'm beginning to think that the problem as stated is false, but can't think of a counterexample. I appreciate any guidance.

REPLY [2 votes]: A convex function on $[0,1]$ that is not continuous (and therefore not absolutely continuous) is
$$ f(x) = \cases{1 & if $x=0$\cr
                 0 & otherwise}$$
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