A continuous function $f$ such that $\frac{\partial f}{\partial x}$ does not exists but $ \frac{\partial^2 f}{\partial x\partial y}$ exists Does there exist a continuous function $f:\mathbb R^2\to \mathbb R$ such that $\displaystyle \frac{\partial f}{\partial x}$ does not exists but $\displaystyle \frac{\partial^2 f}{\partial x\partial y}$ exists.
I think yes. But I am unable to find an example of such function. Can anyone help me ?

See this Wikipedia page on the Weierstrass function $W_{\alpha}(x)$. It is continuous everywhere, but differentiable nowhere. So let
$$f(x,y)=W_{\alpha}(x)$$
Then $\frac{\partial f}{\partial x}$ does not exist anywhere, and since $\frac{\partial f}{\partial y}=0$ the mixed derivative $\frac{\partial^2 f}{\partial x\partial y}$ exists everywhere (and is also equal to 0).
