Given a function taking in two variables defined on some closed subspace of real numbers, when is it true that if we multiply the partial derivatives we get the cross-partial derivative of that function.
That is, when is it true that given given a function $f(x,y)$ we have
$$ \frac{\partial f}{\partial x} \cdot \frac{\partial f}{\partial y} = \frac{\partial f}{\partial x \partial y} $$
If there is no result available that guarantees conditions for equality, are there perhaps conditions under which this expression holds with inequality?