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While for a [[discrete random variable]] one could say that an [[event]] with [[probability]] zero is impossible, this can not be said in the case of a continuous random variable, because then no value would be possible.
 
This [[paradox]] is solved by realizingréalizing that the probability that ''X'' attains a value in an [[uncountable]] set (for example an [[interval (mathematics)|interval]]) can not be found by adding the probabilities for individual values.
 
By another convention, the term "continuous random variable" is reserved for random variables that have [[probability density function]]s. A random variable with the [[Cantor distribution]] is continuous according to the first convention, and according to the second, is neither continuous nor discrete nor a weighted average of continuous and discrete random variables.