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A twice continuously differentiable real function has equal mixed second partial derivatives.
Let be open and be of class . Then for every and all indices ,
Apply Schwarz's theorem at each . The hypothesis guarantees that the mixed partial derivatives exist and are continuous near .
An open set in a topological space is a subset such that .
For a nonnegative integer , a class map is a function defined on an open set such that all partial derivatives of of total order at most exist on and are continuous on (with order meaning itself).
A map is class , or smooth, if it is class for every nonnegative integer .
A mixed partial derivative of a scalar function (with ) at is a second-order partial derivative of the form
provided the relevant partial derivatives exist.
Let be open, , and . The partial derivative of at with respect to the -th coordinate is
when this limit exists in .
Partial derivatives are the entries of the Jacobian matrix. Existence of every partial derivative at does not by itself imply differentiability at .
Let be a function between metric spaces, and let . We say is continuous on if the restriction , with carrying the subspace metric, is continuous. Equivalently, is continuous at every point .
Spelled out: for every and every , there exists such that for all ,