If I have a function $f(x)$ integrated over an infinitesimal slice, where for $\epsilon > 0$ $$\tag{1} \lim_{\epsilon \rightarrow 0} \int_{-\epsilon}^{\epsilon} f(x)dx = 1$$
Then is it true that $f(x)$ can always be expressed in terms of the dirac delta function like so $$f(x) = g(x) + a\delta(x) $$ where $g(x)$ is a function of $x$, $a \in \mathbb{R}$ is a constant, and $\delta(x)$ is the dirac delta function. Or are there other special functions/cases I'm missing, where a function integrated over an infinitesimal slice doesn't vanish?