Derivatives and convexity (Q1194676)

There are proven three theorems. Let \(I\) be the interval \([0,1]\), let \(D\) be the set of all derivatives on \(I\), let \(C_{ap}\) be the set of all approximately continuous functions on \(I\) and \(L\) be the set of all Lebesgue functions on \(I\). Theorem 1. Let \(f=(f_ 1,\dots,f_ n)\), where \(f_ 1,\dots,f_ n\in D\), let \(G\) be an open convex subset of the Euclidean space \(\mathbb{R}^ n\) which contains \(f(I)\) and let \(F\) be a strictly convex function defined on \(G\) such that \(F\circ f\in D\). Then \(f_ 1,\dots,f_ n\) and \(F\circ f\) are in \(L\). Theorem 2. Let \(f\in D\), let \(F\) be a strictly convex function on \(f(I)\) such that \(F\circ f\in D\). Then \(f\in L\). If \(F\) is also continuous, then \(F\circ f\in L\). Theorem 3. Let \(f\) be a non-constant derivative on \(I\) and let \(F\) be a strictly convex function defined on \(f(I)\) such that \(F\circ f\in D\cap C_{ap}\). Then \(F\) is continuous. Theorem 1 is a generalization of the Lemma 4.4 in Proc. Am. Math. Soc. 112, No. 3, 807-817 (1991; Zbl 0746.26002) by the author and \textit{C. E. Weil}. There are some additional remarks to the theorems.