Let's say I have cumulative function of dirac measures $F_{\delta_{a_n}}(x)$ that converge pointwise to $F(x)$ for all $x$ such that $F$ is continuous (which is also a cumulative function).
Now $F_{\delta_{a_n}}(x)=0$ or $F_{\delta_{a_n}}(x)=1$ so that $F(x)=0$ or $F(x)=1.$
Does $F$ have only one discontinuous point ?
Assume the contrary: There exists $x<y$ such that $F$ is discontinuous at $x$ and $y$.
As $F$ is monotone and right-continuous, it follows $F(x) < F(y)$. Since $F$ has countably many discontinuities only, there exists some $c\in (x, y)$ such that $F$ is continuous at $c$. Again, we have $F(c) < F(y)$. Thus, $0 = F(c) \ge F(x) \ge 0$, which contradicts that $F$ is discontinuous at $x$.