# For a continuous random variable, why does $P(a < Z < b) = P(a \leq Z < b) = P(a < Z \leq b) = P(a \leq Z \leq b)$

My textbook puts this in a sidebox with the heading “Note” and doesn’t explain why. Could you tell me why this statement holds?

$P(a < Z < b) = P(a \leq Z < b) = P(a < Z \leq b) = P(a \leq Z \leq b)$

This would explain why $\leq$ and $<$ are basically the same for continuous variables – including or excluding the endpoint really doesn’t change anything – for any point you pick close to the endpoint, there is still an infinite amount of points between them.