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A random variable $X$ is called continuous if its set of possible values is uncountable, and the chance that it takes any particular value is zero ($\text{P}(X = x) = 0$ for every real number $x$). A random variable is continuous if and only if its cumulative probability distribution function is a continuous function.

From Mood et al. (page 60, 1974):

"A random variable $X$ is called continuous if there exists a function $f_{X}(.)$ such that $F_{X}(.)=\int_{-\infty}^{x}f_{X}(u)du$ for every real number $x$. The cumulative distribution function $F_{X}(.)$ of a continuous random variable $X$ is called absolutely continuous".

Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to theory of statistics. (B. C. Harrinson & M. Eichberg, Eds.) (3rd ed., p. 564). McGraw-Hill, Inc.

Excerpt reference: Glossary of Statistical Terms from berkeley.edu

However, the term is also commonly used for variables that can take on a great many values, such as IQ.