Showing that a set is a continuity set Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{B}, P)$. 
Consider $B:=(-\infty,x] \subseteq \mathbb{R}$ with $P(X\leq x)$ continuous at $x$. 
Let $\delta B:=\bar{B}-B^{\circ}$ be the boundary of $B$ where $\bar{B}$ is the closure and $B^{\circ}$ is the interior of $B$. 
Show that $P(X \in \delta B)=0$. 
Any suggestion? In particular, I do not understand what is the boundary of $B$.

Let's use standard notation $\partial$ for boundary. Since $B$ is close set, $\bar{B}=B$ and $B^o=(-\infty, x)$. So $\partial B=\bar{B}-B^o=\{x\}$. Since $P(X\leqslant x)$ is continuous at $x$, 
$$
\lim_{\Delta x\to0}P(X\leqslant x+\Delta x)=P(X\leqslant x)
$$
And thus
\begin{align}
P(X \in \partial B)&=\lim_{\Delta x\to0}P(x<X\leqslant x+\Delta x)
\\
&=\lim_{\Delta x\to0}(P(X\leqslant x+\Delta x)-P(X\leqslant  x))
\\
&=\lim_{\Delta x\to0}P(X\leqslant x+\Delta x)-P(X\leqslant x)
\\
&=0
\end{align}
