Discrete and continuous variables

Discrete variable

Definition: A variable consisting of separate, indivisible categories in which no value exists between neighboring categories.
Example:
1. Counting digits: No other value exists between the 1 and the 2 you get when you roll a die;
2. Different classifications: Classification of personality disorders；
3. Time, length, mass…

Definition: A variable in which there are infinite possible values between any two observations and which can be divided into infinite components.
Expression: Generally it is represented by a continuous straight line of real numbers, on which there are infinite points and you can find an infinite number of points between any two adjacant points.
Real limit: When speaking of an observation of a continuous variable, it often refers to an interval on the real line, and the boundary forming this interval is called the real limit. For instance, 23 represents the range from 22.5 to 23.5, where 22.5 is the lower real limit and 23.5 is the upper real limit.

When the value space of a discrete variable is large and the value points are dense, it can also be regarded as a continuous variable. Grouping continuous variables can be treated as discrete variables.