For continuous distribution (on R) the probability of a single point is $0$. So I'm not sure what does it mean to sample $M$ elements from a continuous distribution.
Let say there is a continuous distribution D and there is a number z and a function f such that: $f(x)=1$ for $x < z$ except of a finite number of cases.
$f(x)=0$ for $x \ge z$ except of a finite number of cases.
$D(x < z) > 0 , D(x \ge z) > 0$
So if I have a random sample $(X_1, \dots ,X_m)$ from $D$, And assume $X_1>z$ , can I conclude that $f(X_1)=0$ ?
And assume $f(X_2)=1$ , can I conclude that $X_2 < z$ ?