I am just reading brownian motion and stochastic calculus by Ioannis Karatzas and Steven Shreve and came up with this problem. Any help would be valuable.
2026-04-09 17:58:40.1775757520
How can I prove that the set of nondifferentiable points of a convex function is countale?
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Sketch of proof. The left and right derivatives $f_-'$, $\,f_+'$ of a convex function $f$ do exist, and they are both increasing, and also $$ f_-(x)\le f_+'(x), \quad\text{for all $x$}. $$ Thus each of them is discontinuous in a countable at most set $S$, and they differ in a subset of $S$.