Points of non-differentiability of convex functions.
Let \(B=\{r_{n}\}_{n=1}^{\infty }\) be a countable subset of the real line. One can easily construct a right continuous nondecreasing function \(g:\mathbb R\rightarrow \mathbb R\) continuous at every \(x\in \mathbb R\backslash B\) and discontinuous at every \(x\in B\) (take, e.g., \(g(x)=\sum_{n:r_{n}\leq x}2^{-n}).\) The graph of \(g\) admits a (unique) maximal (cyclically) monotone extension \(\rho ,\) which is given by \(\rho (x)=[\sup_{y<x}g(y),g(x)].\) There is a convex function \(f:\mathbb R\rightarrow \mathbb R\) whose subdifferential mapping is \(\rho ;\) the set of points at which \(f\) is differentiable coincides with the set of points where \(\rho \) is single-valued, which is obviously \(\mathbb R\backslash B.\) The authors use a different method to construct a convex function differentiable at every \(x\in \mathbb R\backslash B\) and not differentiable at any \(x\in B.\)
scientific article; zbMATH DE number 2032659