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Schwarz–Clairaut theorem

Under continuity, mixed second partial derivatives agree.

Schwarz–Clairaut theorem: Let U⊆RnU\subseteq\mathbb R^n be an and let f:U→Rf:U\to\mathbb R. Fix indices i,j∈{1,…,n}i,j\in\{1,\dots,n\}. If the ∂2f∂xi∂xj\frac{\partial^2 f}{\partial x_i\partial x_j} and ∂2f∂xj∂xi\frac{\partial^2 f}{\partial x_j\partial x_i} exist on a neighborhood of a∈Ua\in U and are continuous at aa, then

∂2f∂xi∂xj(a)=∂2f∂xj∂xi(a).\frac{\partial^2 f}{\partial x_i\partial x_j}(a)=\frac{\partial^2 f}{\partial x_j\partial x_i}(a).
Consequences

This justifies treating the of a sufficiently smooth function as symmetric, and it is a standard hypothesis in second-order local analysis such as the .