An extension of Mazur's theorem on Gateaux differentiability to the class of strongly $\alpha (\cdot )$-paraconvex functions
Volume 172 / 2006
Studia Mathematica 172 (2006), 243-248
MSC: Primary 46G05.
DOI: 10.4064/sm172-3-3
Abstract
Let $(X,\| \cdot \| )$ be a separable real Banach space. Let $f$ be a real-valued strongly $\alpha (\cdot )$-paraconvex function defined on an open convex subset ${\mit \Omega } \subset X$, i.e. such that $$ f(tx+(1-t)y) \leq tf(x)+(1-t)f(y) + \mathop {\rm min}[t,(1-t)] \alpha (\| x-y \| ). $$ Then there is a dense $G_{\delta }$-set $A_{\rm G}\subset {\mit \Omega }$ such that $f$ is Gateaux differentiable at every point of $A_{\rm G} $.
