Consider a well-behaved function $f(x)$ defined on $x\geq0$, and construct a discretized version of it using the Dirac-delta function:
$$ f_{\Delta_x}(x)=\sum\limits_{n=0}^{\infty}f(n\Delta_x)\delta(x - n\Delta_x). $$
If we let $\Delta_x \to 0$ in the above, do we recover $f(x)$?
The following notation might be sloppy, but what I mean is:
$$ \lim_{\Delta_x \to 0} f_{\Delta_x}(x)=f(x) \quad ? $$
Convergence $ \lim_{\Delta_x \to 0} f_{\Delta_x}=f $ is true in the sense of distributions. I would say (vaguely) that you can only recover $f$ as some kind of a weak limit of $f_{\Delta_x}$, not a strong one.
Without trying to make a statement with optimal assumptions, the following holds: if $f$ is continuous, then for any continuous, compactly supported $\phi:(0,\infty)\to\mathbb R$ you have $$ \lim_{\Delta_x \to 0} \langle f_{\Delta_x},\phi\rangle = \langle f,\phi\rangle. $$ Here, formally at least, $\langle f,\phi\rangle=\int_0^\infty f(x)\phi(x)dx$.