Interval Definition

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. For example, the half-open interval between and , where is excluded and is included, is denoted as .

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 written as , where and are the endpoints. For example, the open interval between and is written as and is drawn like this on the number line.

  

• Closed interval: A closed interval includes both its endpoints. It is often written as , where and are the endpoints. For example, the closed interval between and is written as and is drawn like this on the number line.

  

• Half-open interval: A half-open interval, also called a half-closed interval, includes one endpoint but not the other. It can be written as or , depending on which endpoint is included. For example, the half-open interval between and , where is included and is not, is written as and is drawn like this on the number line.

  

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.

Range Definition

The range of a function is the set of all possible output values generated by the function for a given domain. It is a measure of how the output values of a function are distributed and can be used to analyze the behavior of the function and its applications in various contexts.

Domain Definition

The domain of a function is the set of all possible input values for which the function is defined.

Set Definition

A set in mathematics is a collection of distinct objects, considered as an object in its own right. Sets are one of the fundamental concepts in mathematics.