Properties of Hadamard Directional Derivatives: Denjoy-Young-Saks Theorem for Functions on Banach Spaces

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.