# construction of Dirac delta function

The Dirac delta function is notorious in mathematical circles for having no actual as a function. However, a little known secret is that in the domain of nonstandard analysis, the Dirac delta function admits a completely legitimate construction as an actual function. We give this construction here.

Choose any positive infinitesimal $\varepsilon$ and define the hyperreal valued function $\delta:\,^{*}\mathbb{R}\longrightarrow\,^{*}\mathbb{R}$ by

 $\delta(x):=\begin{cases}1/\varepsilon&-\varepsilon/2

We verify that the above function satisfies the required properties of the Dirac delta function. By definition, $\delta(x)=0$ for all nonzero real numbers $x$. Moreover,

 $\int_{-\infty}^{\infty}\delta(x)\ dx=\int_{-\varepsilon/2}^{\varepsilon/2}% \frac{1}{\varepsilon}\ dx=1,$

so the integral property is satisfied. Finally, for any real function $f:\mathbb{R}\longrightarrow\mathbb{R}$, choose an infinitesimal $z>0$ such that $|f(x)-f(0)| for all $|x|<\varepsilon/2$; then

 $\varepsilon\cdot\frac{f(0)-z}{\varepsilon}<\int_{-\infty}^{\infty}\delta(x)f(x% )\ dx<\varepsilon\cdot\frac{f(0)+z}{\varepsilon}$

which implies that $\int_{-\infty}^{\infty}\delta(x)f(x)\ dx$ is within an infinitesimal of $f(0)$, and thus has real part equal to $f(0)$.

Title construction of Dirac delta function ConstructionOfDiracDeltaFunction 2013-03-22 12:35:48 2013-03-22 12:35:48 djao (24) djao (24) 5 djao (24) Derivation msc 34L40 msc 26E35