Schwarz's Theorem (Clairaut's theorem)

Schwarz (Clairaut) Theorem: Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}$. Fix indices $i\neq j$. If the mixed second partial derivatives $\frac{\partial^2 f}{\partial x_i\partial x_j}$ and $\frac{\partial^2 f}{\partial x_j\partial x_i}$ exist on a neighborhood of $a\in U$ and are continuous at $a$, then $\frac{\partial^2 f}{\partial x_i\partial x_j}(a)=\frac{\partial^2 f}{\partial x_j\partial x_i}(a).$

This theorem ensures symmetry of the Hessian matrix under standard smoothness hypotheses and is used throughout multivariable analysis and optimization.

Proof sketch: Reduce to the two-variable case. Consider the increment $f(a_i+h,a_j+k)-f(a_i+h,a_j)-f(a_i,a_j+k)+f(a_i,a_j),$ divide by $hk$, and analyze limits as $(h,k)\to(0,0)$ using the mean value theorem twice and continuity of the mixed partials.