Continuous or discrete variable

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.

Continuous variable

A continuous variable is one 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 $a$ and $b$ with $a,b\in \mathbb {R} ;a\neq b$ is infinite and uncountable.

Methods of calculus are often used in problems in which the variables are continuous, for example in continuous optimization problems.

In statistical theory, the probability distributions of continuous variables can be expressed in terms of probability density functions.

In continuous-time dynamics, the variable time is treated as continuous, and the equation describing the evolution of some variable over time is a differential equation. The instantaneous rate of change is a well-defined concept.

Discrete variable

In contrast, a discrete variable over a particular range of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the nearest other permissible value. The number of permitted values is either finite or countably infinite. Common examples are variables that must be integers, non-negative integers, positive integers, or only the integers 0 and 1.

Methods of calculus do not readily lend themselves to problems involving discrete variables. Examples of problems involving discrete variables include integer programming.

In statistics, the probability distributions of discrete variables can be expressed in terms of probability mass functions.

In discrete time dynamics, the variable time is treated as discrete, and the equation of evolution of some variable over time is called a difference equation.

In econometrics and more generally in regression analysis, sometimes some of the variables being empirically related to each other are 0-1 variables, being permitted to take on only those two values. A variable of this type is called a dummy variable. If the dependent variable is a dummy variable, then logistic regression or probit regression is commonly employed.