Points of non-differentiability of convex functions.
type: Item
instance of: scholarly article
title: Points of non-differentiability of convex functions.
zbMATH Open document ID: 1047.26009
DOI: 10.1016/S0096-3003(02)00932-3
author: Samir Siksek
author: Esam El-Sedy
published in: Applied Mathematics and Computation
publication date: 2004-01-26T00:00:00Z
review text: 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 \(x\in B.\)
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 2032659
zbMATH Keywords: convex functions
zbMATH Keywords: non-differentiability
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
cites work: Convex functions and upper semi-continuous collections
cites work: Q5333460
cites work: Q3954445
description: scientific article; zbMATH DE number 2032659