If Double Integral of a(x, y)h(x, y) vanishes for any C^2 h(x, y) then C^0 a(x, y) vanishes

Theorem
Let $\alpha \left({x, y}\right)$, $h \left({x, y}\right)$ be functions in $\R$.

Let $\alpha \in C^0$ in a closed region $R$ whose boundary is $\Gamma$.

Let $h \in C^2$ in $R$ and $h = 0$ on $\Gamma$.

Let:
 * $\displaystyle \int \int_R \alpha \left({x, y}\right) h \left({x, y}\right) \rd x \rd y = 0$

Then $\alpha \left({x, y}\right)$ vanishes everywhere in $R$.

Proof
$\alpha \left({x, y}\right)$ is nonzero at some point in $R$.

Then $\alpha \left({x, y}\right)$ is also nonzero in some disk $D$ such that:
 * $\left({x - x_0}\right)^2 + \left({y - y_0}\right)^2 \le \epsilon^2$

Suppose:


 * $\left({x, y}\right) = \operatorname {sgn} \left({\alpha \left({x, y}\right)}\right) \left({\epsilon^2 - \left({x - x_0}\right)^2 + \left({y - y_0}\right)^2}\right)^3$

in this disk and $0$ elsewhere.

Thus $h \left({x, y}\right)$ satisfies conditions of the theorem.

However:


 * $\displaystyle \int \int_R \alpha \left({x, y}\right) h \left({x, y}\right) \rd x \rd y = \int \int_D \left\vert{\alpha \left({x, y}\right)}\right\vert \left({\epsilon^2 - \left({x - x_0}\right)^2 + \left({y - y_0}\right)^2}\right)^3 \ge 0$

Hence the result, by Proof by Contradiction.