When do partial derivatives fail to commute?

Question

What are the conditions that a function $f(x,y)$ should satisfy for the partial derivatives $f_{xy}$ and $f_{yx}$ to be equal?

Answer 1

See Here

Clairaut’s Theorem

Suppose that $f$ is defined on a disk $D$ that contains the point $(a,b)$. If the functions $f_{xy}$ and $f_{yx}$ are continuous on this disk then $$f_{xy}(a,b) =f_{yx}(a,b)$$

We can actually restrict ourselves a bit less and let $D$ be any open subset of $\mathbb{R}^2$, which generalizes nicely to

Extended Clairaut’s Theorem

Suppose $f$ is a function of variables defined on an open subset $D$ or $\mathbb{R}^n$. Suppose all mixed partials with each possible number of and combination of differentiations in each input variable exist and are continuous on $D$. Then, all the mixed partials are continuous.