Continuous distribution, continuity questions

Question

A distribution function is by definition a right continuous function. For a continuous random variable however, this distinction is not really necessary because the distribution function of the continuous random variable is itself continuous and thus for all points $[0,1]$ the left limit equals the right limit.

My question is then as follows... For a continuous random variable $X$, is the following true?

$$ P[X<x] ?= P[X \leq x] $$

My reasoning is that the above equation is true because for a continuous random variable, $P[X=x] =0$.

Thanks!

Answer 1

Yes, for a continuous random variable the equality $P(X\leq x)=P(X<x)$ is true exactly for the reason you mentioned: $P(X=x)=0$.

It is useful to keep in mind that if the cumulative distribution functions is defined with the non-strict inequality (which is how it is down in most modern textbooks), then the CDF is right-continuous. However, you could also define the CDF with the strict inequality in which case it would have been left-continuous.