Sequence of Distribution Functions Given a sequence of nondecreasing functions (weakly increasing) $F_n:\mathbb{R} \rightarrow \mathbb{R}$, assume:


*

*Each $F_n$ is right continuous.

*There is a nondecreasing function $F:\mathbb R \rightarrow \mathbb R$.

*$F_n(x) \rightarrow F(x)$ to each continuity point of $F$.
Questions Is $F$ continuous? Is $F$ right continuous? Is $F$ well-defined? That is, is there any ambiguity as to the values of $F(x)$ where $F$ is not continuous at $x$? 
My thoughts
1) No $F$ need not be continuous, counterexamples are plenty.
2) Yes $F$ should be right continuous, but I cannot formalize my argument. My thoughts are that $F$ is monotone, therefore it has countably many discontinuities, and in particular its set of continuities are dense in $\mathbb{R}$. Thus for $x \in \mathbb R$, choose a decreasing sequence $x_k \downarrow x$ such that $F$ is continuous at each $x_k$.  Therefore $F_n(x_k) \xrightarrow{n} F(x_k)$. Since each $F_n$ is right continuous, we have that $F_n(x_k) \xrightarrow{k} F_n(x)$. Perhaps it's false??? Thoughts?
3) Yes $F$ is well defined, if it is right continuous. This follows from the well-known facts about monotone functions: limits always exist. Then a discontinuity $x$ is at most a "left-discontinuity" and the value of the function $F$ can be determined by $\lim_{x_k \downarrow x} F(x_k) \equiv F(x)$.

suppose all $F_n$ and $F$ are the same right-continuous function. Then we modify the value of $F$ at its discontinuous points such that it becomes left-continuous and note the modified function $F'$. $F_n$ and $F'$ together satisfy your assumptions 1,2 and 3 while $F'$ is left-continuous. I think the values at discontinuous points of the limit function is quite arbitray, all we could say is that the limit function has an almost surely equivalent version which is right continuous.
For example: $F(x)=F_{n}(x) = \mathbb{1}_{x \geq 0}$ and $F'(x)= \mathbb{1}_{x>0}$. 
