differentiability of the norm of L^1

Question

Let $N_1$ denote the natural norm of the functional space $L^1(\Omega)$, where $\Omega$ is an open domain of $R^n$: $$ N(y)=\int_\Omega |y(x)| dx $$

I have the following question regarding $N_1$:

1. Is $N_1$ differentiable in $L^1(\Omega)-(0)$?
2. If yes, is the differential Lipschitz?

Answer 1

No. In $L^1(0,2)$, let $f=\chi_{[0,1]}$ and $g=\chi_{[1,2]}$. Then $$||f+hg||=1+|h|.$$