Finding a $C^1$ function $F(x,y)$ that is not of class $C^2$

Question

I am trying to find a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ of class $C^1$, but not of class $C^2$.

Meaning that $\frac{\partial^2 F}{\partial x \partial y}$ won't be equal to $\frac{\partial^2 F}{\partial y \partial x}$.

I've thought of a lot of possible functions, but in each one of them there is only a single singular point in which they are not of class $C^2$.

Can someone help me find such a function?

Thanks in advance.

Answer 1

I have found this kind of function on this web site: http://mathworld.wolfram.com/PartialDerivative.html