Clairaut's theorem

If a function's mixed partials are continuous, the order of differentiation doesn't matter — $f_{xy}=f_{yx}$ .

Clairaut's theorem states that if $f_{xy}$ and $f_{yx}$ are both continuous on an open region, then $f_{xy}=f_{yx}$ on that region. The continuity hypothesis is the entire content of the theorem, not fine print: a function can be built whose mixed partials both exist everywhere but aren't continuous at one point, and $f_{xy}(0,0)\ne f_{yx}(0,0)$ actually happens there. Every function built from polynomials, exponentials, and trig functions has continuous partials of every order, so in practice the failure is a curiosity, not a daily concern.

Examples

For $f(x,y)=x^3y^2-xy$ , both mixed partials equal $6x^2y-1$ , confirming Clairaut's theorem for this polynomial.