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

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