By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. That is equivalent to saying that for random variables X with the distribution in question, Pr[X = a] = 0 for all real numbers a, i.e.: the probability that X attains the value a is zero, for any number a. In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...

While for a discrete probability distribution 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. In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... This article does not cite its references or sources. ...

This paradox is solved by realizing that the probability that X attains a value in an uncountable set (for example an interval) can not be found by adding the probabilities for individual values. Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist. ... In mathematics, an uncountable set is a set which is not countable. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

By another convention, the term "continuous probability distribution" is reserved for distributions that have probability density functions. These are most precisely called absolutely continuous random variables (see Radon–Nikodym theorem). In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... Absolute continuity of real functions In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n... In mathematics, the Radon–Nikodym theorem is a result in functional analysis that states that, given a measure space (X,Σ), if a measure ν is absolutely continuous with respect to another measure μ which is sigma-finite, then there is a measurable function f on X and taking values in [0,∞), such...

A random variable with the Cantor distribution is continuous according to the first convention, but according to the second, it is not (absolutely) continuous. Also, it is not discrete nor a weighted average of discrete and absolutely continuous random variables. The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ...

In practical applications random variables are often either discrete or absolutely continuous.