Reduction to the Identity

Let us show that in the $\varepsilon\to 0$ limit the filter reduces to an almost-identity operation, in the sense that it reproduces in the output function $f_{\varepsilon}(x)$ the input function $f(x)$ almost everywhere. If we consider the well-known relation mentioned in the previous section as a possible definition of the Dirac delta ``function'', as the $\varepsilon\to 0$ limit of the first-order kernel $K_{\varepsilon}^{(1)}\!\left(x-x'\right)$, it becomes clear that we have, for an arbitrary integrable function $f(x)$

$$\begin{eqnarray*} \lim_{\varepsilon\to 0} f_{\varepsilon}(x) & = & \lim_{\va... ... \int_{-\infty}^{\infty}dx'\, \delta(x-x') f\!\left(x'\right). \end{eqnarray*}$$

According to the properties of the delta ``function'', this integral returns the value $f(x)$ at every point where this function is continuous. We therefore have

$$\lim_{\varepsilon\to 0} f_{\varepsilon}(x) = f(x),$$

at every point where $f(x)$ is continuous. Since this may fail at a finite (or at least zero-measure) set of points where $f(x)$ is discontinuous, we say that in the $\varepsilon\to 0$ limit the first-order filter reduces to the identity almost everywhere. We may also say that the filter becomes an almost-identity operation in the limit. What happens at the points of discontinuity of $f(x)$ is discussed in the next section.