Suppose I have a set of functions $(f_\epsilon)$ such that as $\epsilon\to 0$, $f_\epsilon\to F$ s.t.
$F(x)=0$ for $x\neq 0$ and $F(x)=\infty$ for $x=0$;
$\int_{-\infty}^\infty f_\epsilon(x) dx=1$ for all $\epsilon >0$
Then can I conclude that the limit is the delta function $\delta(x)$? (which has the sampling property too)
Thanks