An interval is a set of real numbers that lies between two given numbers. These two numbers are often called the endpoints of the interval. An interval can be represented on a number line.
There are several types of intervals, based on whether or not they include their endpoints:
- Open interval: An open interval does not include its endpoints. It is often denoted as (a, b), where a and b are the endpoints.
- Closed interval: A closed interval includes both its endpoints. It is often denoted as [a, b], where a and b are the endpoints.
- Half-open interval: A half-open interval, also called a half-closed interval, includes one endpoint but not the other. It can be denoted as [a, b) or (a, b], depending on which endpoint is included.
Intervals are a fundamental concept in calculus and real analysis. They are used to define important concepts such as limits, continuity, and the derivative. They are also used in the definition of the definite integral, where the interval represents the region over which a function is integrated.