Existence of the partial derivatives ${\delta^{2}f}\over {\delta x \delta y}$ and ${\delta f}\over {\delta x}$

Question

The question is can the partial double derivative ${\delta^{2}f}\over {\delta x \delta y}$ exist without the derivative ${\delta f}\over {\delta x}$ existing?

I don't know , I am new to the multi variable calculus scene and totally clueless about it. Would be really helpful if some explanation is given as to why this may or may not happen

Thanks.

Answer 1

If $f(x,y)=|x|$, then $\frac{\partial f}{\partial x}(0,y)$ does not exist for any $y$. Therefore certainly $\frac{\partial^2 f}{\partial y\partial x}(0,y)$ does not exist, because it does not make sense to take the derivative of a function which is not even defined. On the other hand, $\frac{\partial f}{\partial y}(x,y)=0$ for all $(x,y)$, and thus $\frac{\partial^2 f}{\partial x\partial y}(x,y)=0$ for all $(x,y)$. The bottom line is that for functions which are not sufficiently smooth, the order of partial differentiation matters.