Differentiability of Norms

Let f be a real-valued function on an open subset U of a Banach space X. Let x ∈ U. We say that f is Gâteaux differentiable at x if there is F ∈ X* such that $$\mathop {\lim }\limits_{t \to 0} \frac{{f(x + th) - f(x)}}{t} = F(h)$$ for every h ∈ X.