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 $\delta:\,^*\R \longrightarrow\,^*\R$ 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 $f: \R \longrightarrow \R$ , 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 $f(0)$ .