In the book Fabian and others I saw exercise:
"Let $\|$.$\|_{\infty}$ denote the canonical of $l_{\infty}$ and set $p(x) = \limsup |x_i|$. Define $\||x\|| = \|x\|_{\infty} + p(x)$ for $x \in l_{\infty}$. Show that norm $\||$.$ \||$ is nowhere Gateaux differentiable."
I can prove this statement in several points. I know where the norm $\|$.$\|_{\infty}$ is Gateaux differentiable. It is known that the norm of $l_{\infty}$ is Gateaux differentiable on a dense set. But I don't know how do the other points.
Please, can you say what do in this case.
Thanks for your time.