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 $\epsilon$ and define the hyperreal valued function $\delta {:}^{*}ℝ{⟶}^{*}ℝ$ by

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,

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)𝑑x={\int }_{-\epsilon /2}^{\epsilon /2}\frac{1}{\epsilon }𝑑x=1,$$

so the integral property is satisfied. Finally, for any continuous real function $f:ℝ⟶ℝ$, choose an infinitesimal $z>0$ such that for all ; then

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