A note on maximum-type functions

The paper studies the differential properties of functions of the form

$g(x) = \mathop {\max }\limits_{y \in Y} f(x,y),$

where x ∈ X (X is an open convex set from ? ^m ) and y ∈ Y (Y is a compact from ? ⁿ ). Apart from the conventional smoothness conditions imposed on f(x, y), the condition of the concavity of g(x) on X is also imposed.

The differentiability of function g(x) on X is proved.The results of the study facilitate the derivation of the conditions ensuring the sufficiency of Pontryagin’s maximum principle.