Continuous distribution, continuity questions

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!

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