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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?

stochasticmrfox
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    For what it's worth, CDFs can be defined to be left-continuous instead of right-continuous. See https://math.stackexchange.com/questions/1807120/why-arent-cdfs-left-continuous, https://math.stackexchange.com/questions/3221379/why-does-a-c-d-f-need-to-be-right-continuous?noredirect=1&lq=1 – StubbornAtom Dec 28 '19 at 13:43

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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 $$\lim_{x \to a^{+}} F_{X}(x)=\lim_{x \to a^{-}} F_{X}(x)+P(X=a)=P(X\leq a)$$

kludg
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