Mixed Partial Derivative of Heaviside Step Function

Theorem
Let $\tuple {x, y} \stackrel u {\longrightarrow} \map u {x, y}: \R^2 \to \R$ be the Heaviside step function.

Let $u := T_u$ be the distribution associated with $u$.

Let $\delta_{\tuple {0, 0} } \in \map {\DD'} {\R^2}$ be the Dirac delta distribution.

Then in the distributional sense:


 * $\dfrac {\partial^2 u} {\partial x \partial y} = \delta_{\tuple {0, 0}}$

Proof
Let $\phi \in \map \DD {\R^2}$ be a test function with support on $\openint 0 a^2 := \openint 0 a \times \openint 0 a$ where $\times$ is the Cartesian product and $a > 0$.

Then: