My professor just went over weak convergence in lecture and mentioned that there is a continuity requirement for the points of F which we require to converge. He offered this example as a way to see why this is required but I am having trouble working it through.
Consider a sequence of random variables with the pmf's $P(X_n = 1 + \frac{1}{n}) = 1, n \geq 1$, and $P(X = 1) = 1$, and let $F_n$ and $F$ be the associated cdf's. Show that $F_n => F$.
We have $F_n(t)=1$ for $t\geqslant 1+1/n$ and $F_n(t)=0$ for $t\lt 1+1/n$. Therefore, if $t\leqslant 1$, $F_n(t)=0$ for any $n$ and $F_n(t)\to 0$, and if $t\gt 1$, then $F_n(t)=1$ for $n$ large enough hence $F_n(t)\to 1$.
The function $F$ is continuous everywhere, except at $1$. We have $F_n(1)=0$ for any $n$ but $F(1)=1$. Since $F_n(t)\to F(t)$ for any $t\neq 1$, we have the convergence in distribution of $F_n$ to $F$.
This examples shows that the convergence $F_n(t)\to F(t)$ may fail at points where $F$ is not continuous, even when we have weak convergence.