Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces

Abstract

The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f:\mathbb{R}\to \mathbb{R}$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on ${\mathbb{R}}^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f_H^{+}(x,v)$, $f_H^{-}(x,v)$ ($v\in X$) which holds almost everywhere for an arbitrary function $f:{\mathbb{R}}^2\to \mathbb{R}$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.