Example of a function of two variables that has partial derivatives but the mixed derivative does not exist

Question

I'm working on an exercise on harmonic functions and stumbling upon finding a function of two variables that has partial derivatives with respect to each variable but the mixed derivative does not exist.

To be specific, I want to find a real-valued function of two real variables $x$ and $y$, $f(x, y)$ that satisfies both partial derivatives $f_{x}$ and $f_{y}$ each exist on an open interval $\mathit{I}$ but at least $f_{xy}$ or $f_{yx}$ does not exist on that interval.

Answer 1

Take $$g(t)= \begin{cases} t^2, t\geq 0\\ -t^2, t<0 \end{cases}$$ It is easy to see that $g'$ exists everywhere but $g''$ exists everywhere except $0$. Now take for example $f(x,y)=g(x+y)$.