By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. That is equivalent to saying that 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.
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 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.
By another convention, the term "continuous random variable" is reserved for random variables that have probability density functions. 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.
In practical applications random variables are often either discrete or continuous.