Dirac delta function

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

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

For any continuous function $F$:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)F(x)𝑑x=F(0)$$

or in $n$ dimensions:

$${\int }_{{ℝ}^n}\delta (x-s)f(s)d^{n}s=f(x)$$

$\delta (x)$ 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 availablehttp://www.rose-hulman.edu/ rickert/Classes/ma222/Wint0102/dirac.pdfon line.

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