showing weak convergence from assumption Hoi, if $\mu_1,\mu_2,\cdots$ are probability measures on $(\mathbb{R}, \mathcal{B}),$
with distribution functions $F_1,F_2,\cdots $
I want to show: If for all $x\in C_F$ (all $x$ where $F$ is continuous), we have $F_n(x)\to F(x)$
then we have weak convergence $\mu_n \to^w \mu$, which is equivalent with
$\mu_n(f)\to \mu(f)$ for all continuous $f\in \cal C_b(\mathbb{R})$
I've been suggested the following procedure:

*

*Let $\epsilon> 0$, and let $K$ s.t. $F(K)-F(-K)>1-\epsilon$


*Approximate a function $f\in \cal C_b(\mathbb{R})$ with piecewise constant function.


*Compute integrals of this approximating function and use convergence of the $F_n(x)$ at continuity points of $F$.
But can't we just say that the set of discontinuities of all $F_n$ and $F$ are at most countable? Then for any interval $I=[x,y]$ where $x,y$ are no discontinuities of any $F_n,F$,  conclude that $\mu_n(\textbf{1}_If)\to \mu(\textbf{1}_I f)$
Second, I'm not sure how to approximate this $f$..
thanks in advance for any insight
 A: The idea is the following. Choose $K_1$ and $K_2$ such that $F(K_2)-F(K_1)\geqslant 1-\varepsilon$ and $K_1,K_2$ are not discontinuity points of $F$. For any integer $N$, find an integer $J_N$ and $\{t_N^k\}_{k=1}^{J_N}$ such that $K_1=t_N^1<t_N^2<\dots<t_N^{J_N-1}<t_N^{J_N}=K_2$, $t_N^j$ are not discontinuity points of $F$ and $\max_{1\leqslant j\leqslant K_N-1}|t_N^{j+1}-t_N^j|\leqslant N^{-1}$. This is possible, as the discontinuity points of $F$ form an at most countable set. Then define 
$$f_N(x)=\sum_{j=1}^{J_N-1}f(t_N^j)\chi_{(t_N^j,t_N^{j+1}]}(x).$$
We have 
\begin{align}
\left|\int_{\Bbb R}f(x)d\mu_n-\int_{\Bbb R}f(x)d\mu\right|&\leqslant \int_{[K_1,K_2]}|f(x)-f_N(x)|d\mu_n+\int_{[K_1,K_2]}|f(x)-f_N(x)|d\mu\\
&+\varepsilon\lVert f\rVert_{\infty}+\left|\int_{(K_1,K_2]}f_N(x)d\mu_n-\int_{(K_1,K_2]}f_N(x)d\mu\right|\\
&\leqslant 2\omega(f_{\mid [K_1,K_2]},N^{-1})+\varepsilon\lVert f\rVert_{\infty}+\sum_{j=1}^{J_N-1}|F_n(t_N^{j+1})-F(t_n^j)|,
\end{align}
where $\omega(g,\delta):=\sup\{|g(x)-g(y)|,|x-y|\leqslant\delta\}$ is the modulus of continuity of $g$. 
This gives for all $N$ that 
$$\limsup_{n\to+\infty}\left|\int_{\Bbb R}f(x)d\mu_n-\int_{\Bbb R}f(x)d\mu\right|\leqslant 2\omega(f_{\mid [K_1,K_2]},N^{-1})+\varepsilon\lVert f\rVert_{\infty}.$$
Now use uniform continuity of $f$ on the compact $[K_1,K_2]$ and let $N\to +\infty$ to get convergence in law. 
