Does there exist any continuous function whose partials doesn't exist? Does there exist a continuous function of $f : \mathbb R^2 \longrightarrow \mathbb R$ such that it is continuous whose both the partial derivatives don't exist.
I think the function $f : \mathbb R^2 \longrightarrow \mathbb R$ defined by $f(x,y) = |x|(1 + y)$, where $(x,y) \in \mathbb R^2$ has the above property at $(0,0)$. But I can't prove that $f$ is continuous at $(0,0)$ by $\epsilon-\delta$ method. Please help me.
Thank you in advance.

Just consider the sum of two one-variable Weierstrass functions, one in the $x$ variable, one in the $y$ variable. This is even better, it has continuity everywhere and no differentiability or partial derivatives anywhere.
