Let's consider a continuous random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$.

Is there a meaningful sense in which one could say that for any $x_0:p(x_0)=0$ the event $X=x_0$ is impossible? By impossible, I mean that if we simulate $X$ then we never observe in the sample the value $x_0$.

If the answer is no, how do we make sure that $X$ will never get the value $x_0$?

Note that my question is complementary to the well-known fact that any event of the form $X=x_0$ is almost impossible, in the sense that $\mathbb{P}(X=x_0)=0$, and this not means that such events cannot happen (i.e., if we simulate $X$ then we can observe the value $x_0$ in the sample).