Partial Differentiation Operator is Commutative for Continuous Functions

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Theorem

Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$.

Let $\map f {x, y}$ be continuous.

Let the partial deriviatives of $f$ also be continuous.


Then:

$\dfrac {\partial^2 f} {\partial x \partial y} = \dfrac {\partial^2 f} {\partial y \partial x}$


Proof


Sources