The Dirac delta ``function'' $\delta(x)$ , or distribution is not a true function because it is not uniquely defined for all values of the argument $x$ . Similar to the Kronecker delta symbol, the notation $\delta(x)$ stands for

$$ \delta(x) = 0 \;\text{for}\; x \ne 0, \;\text{and}\; \int_{-\infty}^\infty \delta(x) dx = 1 $$

For any continuous function $F$ :

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

or in $n$ dimensions:

$$\int_{\mathbb{R}^n} \delta(x - s)f(s) \, d^ns = f(x)$$

$\delta(x)$ can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width.

Notes: However, the limit of the normalized Gaussian function is still meaningless as a function, but some people still write such a limit as being equal to the Dirac distribution considered above in the first paragraph.
An example of how the Dirac distribution arises in a physical, classical context is available on line.

Remarks: Distributions play important roles in Dirac's formulation of quantum mechanics.

Bibliography

1

W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

2

L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.

3

Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)