Let $X, Y$ be Banach spaces. A mapping $F: X\rightarrow Y$ is said to be Gateaux differentiable at $x_0\in X$ iff there exists a continuous linear mapping $A: X\rightarrow Y$ such that $ \textbf{(*)} \quad \lim_{t\downarrow 0}\frac{F(x_0+th)-F(x_0)}{t}=A(h) \quad \forall h\in X. $ I would like to construct a nonlinear mapping $F: X\rightarrow Y$ that is not Gateaux differentiable at $x_0\in X$ but there exists a discontinuous linear mapping $A: X\rightarrow Y$ such that (*) is satisfied.
Thank you for all comments and helping.