DOI: 10.68381/jca22009 ISSN: 0944-6532
Properties of Hadamard Directional Derivatives: Denjoy-Young-Saks Theorem for Functions on Banach Spaces
Luděk Zajíček
The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on
\mathbb{R}^2
R
2
. This extension gives the strongest relation among upper and lower Hadamard directional derivatives
f^+_H (x,v)
f
H
+
(
x
,
v
)
,
f^-_H (x,v)
f
H
−
(
x
,
v
)
(
v \in X
v
∈
X
) which holds almost everywhere for an arbitrary function
f:\mathbb{R}^2\to \mathbb{R}
f
:
R
2
→
R
. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.