Here is a quote from Calculus: Early Transcendentals written by James Stewart
Clairaut’s Theorem:Suppose $f$ is defined on a disk $D$ that contains the point $(a, b)$. If the functions $f_{xy}$ and $f_{yx}$ are both continuous on $D$, then $f_{xy} = f_{yx}$
Where $$f_{xy} \equiv \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right)$$
I don't know how to compose a function that $f_{xy}$ or $f_{yx}$ are not continuous on $(a, b)$. Can you give me an example?