Integral of Dirac Delta Function over Reals

Theorem
Let $\map \delta x$ denote the Dirac delta function.

Then:


 * $\ds \int_{-\infty}^{+\infty} \map \delta x \rd x = 1$

Proof
We have that:


 * $\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$

where:


 * $\map {F_\epsilon} x = \begin {cases} 0 & : x < -\epsilon \\ \dfrac 1 {2 \epsilon} & : -\epsilon \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$

We have that:

Then:

Hence the result.