I've learned that function $f:\mathbb{R}^n\rightarrow \mathbb{R}^p$, at $x \in \mathbb {R}^n$, $f(x)$ is Gateaux differentiable, then exits a linear operator $\lambda (x):\mathbb{R}^n\rightarrow \mathbb{R}^p$ such that $D_uf(x)=\lim_{t \rightarrow 0}{\frac{f(x+tu)-f(x)}{t}}=\lambda(x)(u),\forall u \in \mathbb{R}^n$. If $\lambda(x)$ is continous, $f(x)$ is Frechet differentiable.
I want to find a counterexample that $f(x)$ is Gateaux differentiable and $\lambda(x)$ is linear, not continous, but it's hard for me, help me, thank you!