Differential mean value properties for quasi-continuous functions (Q2762826)

Let \(X\) be a real locally convex space and let \(f: \mathbb{R}\rightarrow X\) be a function such that \(F(t-0)\) exists for every \(t\in(a,b)\) and the left derivative of \(F\) exists everywhere, except for a countable subset \(A\) of \((a,b)\). The paper under review aims to indicate the relationship between \((F(b)-F(a)/(b-a)\) and the convex hull of the set \(\{d^-F(t)\mid t\in(a,b)\}\), where \(B\) is a Lebesgue null set including \(A\). In this respect, it extends a resut due to \textit{R. M. McLeod} [Proc. Edinb. Math. Soc., II. Ser. 14, 197-209 (1965; Zbl 0135.34301)].