Continuous and discrete variables

In mathematics, a variable may be continuous or discrete. If it can take on two particular real values such that it can also take on all real values between them (even values that are arbitrarily close together), the variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value. In some contexts a variable can be discrete in some ranges of the number line and continuous in others.

A continuous variable is a variable which can take on infinitely many, uncountable values.

For example, a variable over a non-empty range of the real numbers is continuous, if it can take on any value in that range. The reason is that any range of real numbers between ${\displaystyle a}$ and ${\displaystyle b}$ with ${\displaystyle a,b\in \mathbb {R} ;a\neq b}$ is infinite and uncountable.

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