Riemann Integrable Dirac Function does not Exist

Theorem
Let $\delta : \R \to \R$ be a real function.

Let $\phi : \R \to \R$ be a smooth function vanishing outside $\closedint a b$.

Let $a \in \R_{> 0}$ be a real number.

Suppose $\delta$ is Riemann integrable on $\closedint {-a} a$.

Suppose for every $\phi$ we have that:


 * $\ds \int_{-a}^a \map \delta x \map \phi x = \map \phi 0$

Then $\delta$ does not exist.

Proof
$\delta$ exists.

Let $\phi$ be a test function of the following form:


 * $\map \phi x = \begin {cases}

\map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$

Let $n \in \N$ be a natural number.

Let $\phi_n : \R \to \R$ be a real function such that:


 * $\forall n \in N : \forall x \in \R : \map {\phi_n} x := \map \phi {n x}$

Then $\map \phi {nx}$ is smooth, vanishes outside $\ds \closedint {- \frac 1 n} {\frac 1 n}$ and:


 * $\forall x \in \R : 0 \le \map \phi {nx} \le 1$

Hence:

This is a contradiction.