Suppose I have a random variable $X$, and I also have $w \in \Omega$, where $\Omega$ is the sample space of the [probability space](https://en.wikipedia.org/wiki/Probability_space). What is the meaning of $X(w)$, is it a random variable also? I unfortunately have no further information regarding $X$ or $w$ than this. 

1. What if $w \not\in \Omega$? Then does $X(w)$ make sense?
2. Does the interpretation of $X(w)$ change if $X$ is a discrete random variable versus continuous?