Schwarz's theorem for C^2 functions

Statement

Let $U \subseteq \mathbb{R}^n$ be open and let $f \colon U \to \mathbb{R}$ be of class C^2 on $U$. Then for all $i, j \in \{1, \dots, n\}$ and each $a \in U$, $$\frac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \frac{\partial^2 f}{\partial x_j \, \partial x_i}(a).$$

Remarks

A function of class $C^2$ is twice differentiable at each point, so this is a case of the symmetry of second derivatives. The result is also called Clairaut's theorem. By induction, for a function of class $C^k$ the partial derivatives of order at most $k$ do not depend on the order of differentiation. So for $C^k$ functions one can use multi-index notation $\partial^\alpha f$ without ambiguity.

Explanations

L1 human

If the second partial derivatives are continuous, then $f_{xy} = f_{yx}$. This saves work: in a $C^2$ function of $n$ variables, only $n(n+1)/2$ of the $n^2$ second partial derivatives can differ.