Let $\mu$, a finite measure and $F$ increasing,right continuous function which is defined as $F(x)=\mu((-\infty,x])$.
We also know that $\mu$ is continuous at a single point $x$ if $\mu(\left\{x\right\})=0$.
I would like to show that $F(x)$ is continuous at x if and only if $\mu(\left\{x\right\})=0$.
Any idea on how to start it ??
On
Try the following: Let $r>0$ and $x\in\mathbb{R}$, then $$F(x)-F(x-r)=\mu((-\infty,x])-\mu((-\infty,x-r])=\mu((x-r,x]).$$ Now $F$ is continuous at $x$, if $\lim_{r\searrow 0}F(x-r)=F(x)$, hence you need to show $$\lim_{r\searrow 0}\mu((x-r,x])=\mu(\{x\}).$$ This can be done by standard continuity of a measure, see https://en.wikipedia.org/wiki/Measure_(mathematics)#Continuity_from_above
Observe that $\{x\}=\bigcap_{n=1}^{\infty}(x-\frac1n,x]$ and prove that: $$\mu(\{x\})=\lim_{n\to\infty}\mu((x-\frac1n,x])$$
Further note that $\mu((x-\frac1n,x]=F(x)-F(x-\frac1n$) so the RHS of $(1)$ takes value $0$ iff (the non-decreasing function) $F$ is left continuous at $x$.
We allready know that $F$ is right continuous at $x$ so the condition of being left continuous at $x$ is the same as the condition of being continuous at $x$.