TITLE: Dirac delta sequences
QUESTION [7 upvotes]: Is it true that any sequence of real functions $(\delta_n)_n$, such that
$$\lim_{n\to\infty} \delta_n(x) = 0 \qquad \forall\,x\ne 0$$
and 
$$\int_{-\infty}^\infty \delta_n(x)\,dx = 1 \ ,$$
tends to a delta function,
$$\lim_{n\to\infty} \delta_n(x) = \delta(x)$$
in a sense that
$$\lim_{n\to\infty} \int_{-\infty}^\infty \phi(x)\,\delta_n(x) \, dx = \phi(0)$$
for any real test function $\phi$? If no, what else should one assume so that the sequence $(\delta_n)_n$ necessarly tends to a delta function?

REPLY [2 votes]: You could set have
$
\delta_n(x) = n/2
$
if $x\in (0,1/n]$ or $x\in (1,1+1/n]$.
Then
$$
\lim \int_{-\infty}^\infty \phi(x)\delta_n(x)dx = \frac12(\phi(0)+\phi(1))
$$
for all continuous functions $\phi$.