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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.

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Can we cheat taking $F$ linear and discontinuous? – Davide Giraudo Dec 9 '12 at 16:06
Davide Giraudo: Yeah. You are right. How about $F$ is nonlinear? Thank you for your comment. – blindman Dec 9 '12 at 16:10
And with $F^2$, where $F$ is linear and discontinuous? – Davide Giraudo Dec 9 '12 at 18:07
up vote 3 down vote accepted

Take $X$ an infinite dimensional Banach space, and $Y:=\Bbb R$. There exists a non-continuous linear functional $f$. Let $F(x):=f(x)^2$ and let $x_0$ such that $f(x_0)\neq 0$. Then \begin{align} F(x_0+th)-F(x_0)&=(f(x_0)+tf(h))^2-f(x_0)^2\\ &=2tf(x_0)f(h)+t^2f(x_0)^2, \end{align} so the $A$ which would work is $A(h):=2f(x_0)f(h)$, which is not continuous.

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