A linear functional δ mapping the compactly supported continuous real functions to their value at zero argument: $\begin{array}{l}\delta: \, C^0 \rightarrow \mathbb{R}\end{array}$ though the following integral:

(1)	$\begin{array}{l}\displaystyle f(0) = \int_{-\infty}^{-\infty} f(x) \, \delta (x) \, dx\end{array}$

which also means that:

(2)	$\begin{array}{l}\displaystyle \int_{-\infty}^{-\infty} \delta (x) \, dx = 1\end{array}$

(3)	$\begin{array}{l}\delta (x) = \begin{cases} + \infty, & \mbox{if } x = 0 \\ 0, & \mbox{if } x \neq 0 \end{cases}\end{array}$

(see schematic representation on Fig. 1)

Fig. 1. Schematic representation of Dirac Delta Function

The base definition of (1) can be naturally generalized to the wider class of real functions over the n-dimensional space $\begin{array}{l}C^0_n: \mathbb{R}^n \rightarrow \mathbb{R}\end{array}$ as:

(4)	$\begin{array}{l}\displaystyle f(0) = \int_{\mathbb{R}^n} f({\bf r}) \, \delta ({\bf r}) \, dV\end{array}$

where

$\begin{array}{l}{\bf r} = (x^1, ... , x^n)\end{array}$	position vector in $\begin{array}{l}\mathbb{R}^n\end{array}$
$\begin{array}{l}dV = dx^1 ... dx^n\end{array}$	infinitesimal volume element in $\begin{array}{l}\mathbb{R}^n\end{array}$

The above definition (4) can be thought as a product of 1-dimensional delta functions:

(5)	$\begin{array}{l}\displaystyle \delta ({\bf r})= \delta (x^1) ... \delta (x^n)\end{array}$

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Dirac Delta Function = δ

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