Geodesic
Properties of Hadamard Directional Derivatives: Denjoy-Young-Saks Theorem for Functions on Banach Spaces
Journal of convex analysis, Tome 22 (2015) no. 1, pp. 161-176
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\newcommand{\R}{{\mathbb R}} The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\R^2$. This extension gives the 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 $f:\R^2\to \R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.
Classification : 46G05, 26B05
Mots-clés : Hadamard upper and lower directional derivatives, Denjoy-Young-Saks theorem, separable Banach space, Hadamard differentiability, Frechet differentiability, Hadamard subdifferentiability, Frechet subdifferentiability, Gamma-null set, Aronszajn null set 
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     author = {L. Zaj{\'\i}cek},
     title = {Properties of {Hadamard} {Directional} {Derivatives:} {Denjoy-Young-Saks} {Theorem} for {Functions} on {Banach} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {161--176},
     year = {2015},
     volume = {22},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a8/}
}
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L. Zajícek. Properties of Hadamard Directional Derivatives: Denjoy-Young-Saks Theorem for Functions on Banach Spaces. Journal of convex analysis, Tome 22 (2015) no. 1, pp. 161-176. http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a8/