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# construction of Dirac delta function

The Dirac delta function is notorious in mathematical circles for having no actual realization 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

$\delta(x):=\begin{cases}1/\varepsilon&-\varepsilon/2<x<\varepsilon/2,\\ 0&\text{otherwise.}\end{cases}$ |

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 continuous real function $f:\mathbb{R}\longrightarrow\mathbb{R}$, choose an infinitesimal $z>0$ such that $|f(x)-f(0)|<z$ 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 $f(0)$.

## Mathematics Subject Classification

34L40*no label found*26E35

*no label found*

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