Gâteaux differentiable functions are somewhere Fréchet differentiable
Generalized fatou inequalities
Generalized first class selectors for upper semi-continuous set-valued maps in Banach spaces
In this paper we deal with weakly upper semi-continuous set-valued maps, taking arbitrary non-empty values, from a non-metric domain to a Banach space. We obtain selectors having the point of continuity property relative to the norm topology for a large class of compact spaces as a domain. Exact conditions under which the selector is of the first Borel class are also investigated.