If two-variable function $\ f(x,y)$ is partialy differentiable respect to both variables ; if both $\frac{\partial f}{\partial x} $ and $\frac{\partial f}{\partial y} $ exist, is it always true the second mixed partial $\frac{\partial^2 f}{\partial x \partial y} $ exist?
If it's not the general case, what could be the weakest condition that fit?
Well using my intuition $\ f_x$ and $\ f_y$ being continuous is the best I could imagine, but I guess that's not all? Please enlighten me.
I'm currently learning calculus by Stewart calculus ed.8
If exact answer requires rigorous understanding of math concepts Which books/texts do I need to look for?
Thanks