Let $F$ denote a distribution function of a probability measure $P$ on a probability space $(\mathbb{R},\mathcal{B})$, where $\mathcal{B}$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$.
Given the definition $$F(x):=P\left((-\infty,x)\right)\;,\;\;\;x\in\mathbb{R}\;,$$ I want to express $P\left([a,\infty )\right)$ in terms of $F$. From a logical point of view, it must hold $$P\left([a,\infty )\right)=P(\mathbb{R})-P\left((-\infty,a])\right)+P\left(\left\{a\right\}\right)=1-F(a)+\text{?}$$ So, the real question is: How can we express $P\left(\left\{a\right\}\right)$ in terms of $F$?
By definition, $$F(x)=P\left((-\infty,x]\right)\;,\;\;\;x\in\mathbb{R}$$
Now, $P(\{a\}) = \lim_{n \to \infty} P\left( (a - 1/n, a] \right) = F(a) - F(a-)$
where $F(a-)$ is the left limit of $F$ at $a$.