# A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet differentiable?

Fr\'echet differentiability implies G\^ateaux differentiability, but the converse is true only for finite-dimensional Banach spaces, in general. As an example, the mapping $f:L^1[0,\pi]\rightarrow\mathbb{R}$ defined by $f(x)=\int_0^\pi\mbox{sin}(x(t))dt$ is every where G^ateaux differentiable, but nowhere Fr\'echet differentiable [9]
• Notice that non separability of $L_1^*$ is essentially for the example user46855 gave. Preiss' differentiability theorem says that every real valued Lipschitz function on a space with separable dual has a point of Frechet differentiability. – Bill Johnson Mar 25 '14 at 14:57