Geodesic
An extension of Mazur's theorem on Gateaux differentiability to the class of strongly $\alpha (\cdot )$-paraconvex functions
S. Rolewicz 1
1 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland
Studia Mathematica, Tome 172 (2006) no. 3, pp. 243-248 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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} $.
DOI : 10.4064/sm172-3-3
Keywords: cdot separable real banach space real valued strongly alpha cdot paraconvex function defined convex subset mit omega subset t leq t mathop min t alpha x y there dense delta set subset mit omega gateaux differentiable every point

S. Rolewicz  1

1 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland 
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S. Rolewicz. An extension of Mazur's theorem
 on Gateaux differentiability
 to the class of strongly $\alpha (\cdot )$-paraconvex functions. Studia Mathematica, Tome 172 (2006) no. 3, pp. 243-248. doi: 10.4064/sm172-3-3

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