On intermediate differentiability of Lipschitz functions on certain Banach spaces

Abstract:

A real-valued function $f$ defined on a Banach space $X$ is said to be intermediately differentiable at $x \in X$ if there is $\xi \in {X^*}$ such that for every $h \in X$ the value $\left \langle {\xi ,h} \right \rangle$ lies between the upper and lower derivatives of $f$ at $x$ in the direction $h$. We show that if $Y$ contains a dense continuous linear image of an Asplund space and $X$ is a subspace of $Y$, then every locally Lipschitz function on $X$ is generically intermediately differentiable.