# Proving $\int_\limits{\infty}^{\infty}\delta(x)dx=1$

From Wikipedia I found this about the delta-function

$$\delta(x)=\begin{cases}\infty,\:\:x=0\\0,\:\:x\neq 0\end{cases}$$

$$\int_\limits{-\infty}^{\infty}\delta(x)dx=1$$

I tried to prove that $$\int_\limits{-\infty}^{\infty}\delta(x)dx=1$$.

$\int_\limits{-\infty}^{\infty}\delta(x)dx=\int_\limits{-\infty}^{0}\delta(x)+\int_\limits{0}^{+\infty}\delta(x)$

However, I do not see how these integrals are going to yield $1$ as a result.

Question:

How do I prove $\int_\limits{-\infty}^{\infty}\delta(x)dx=1$?