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 of one variable was extended by U.S. Haslam-Jones (1932) and A.J.

Ward (1935) to arbitrary functions of two variables. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.

Keywords

• 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