What's the importance of CDFs being right continuous?

Question

One of the properties of a cumulative distribution function $F_{X}(x)$ is that it's right continuous i.e. $$\lim_{x \to a^{+}} F_{X}(x)= F_X(a) \space \forall a \in \mathbb{R}$$.

1)What is the importance of this?

2)Why are CDFs not left continuous?

Answer 1

It follows from the definition of CDF: $$F_X(x)=P(X\leq x)$$ If $P(X=a)>0$ so $F_X(x)$ is discontinuous at $x=a$, then $$\lim_{x \to a^{-}} F_{X}(x)=P(X<a)$$ and $$\lim_{x \to a^{+}} F_{X}(x)=\lim_{x \to a^{-}} F_{X}(x)+P(X=a)=P(X\leq a)$$