Does there exist a function $f$ such that $∂^2f/∂x\,∂y$ exist but $∂f/∂x$ does not exist?

Question

Does there exist a function $f(x,y)$ such that $∂^2f/∂x\,∂y$ exist but $∂f/∂x$ does not exist? I can't find any such $f$.

Answer 1

Take this with a heavy grain of salt because I'm not completely sure my answer is valid, but you can look at $f(x)=|x|e^{|x|y}$. If you try to look at $\frac{∂f}{∂x}$ at the point $(0, 0)$ you see that it doesn't exist. But $\frac{∂^2f}{∂x\,∂y}$ at that point is $0$.

Answer 2

$∂^2f/∂x\,∂y$ by definition is $$\frac {\partial}{\partial x}( \frac {\partial f}{\partial y})$$

If you have a function where $\frac {\partial f}{\partial y}$ is constant while$\frac {\partial f}{\partial x}$ does not exist, then you have a positive answer.

For example $$f(x,y)=y+|x|$$ at $(0,0)$ is such a function.