# Domain of a Function | Open or Closed Interval?

• Niles
In summary, the domain of a function refers to the set of all possible input values for a given function. An open interval is a range of values that does not include the endpoints, while a closed interval includes the endpoints. The type of interval for a function depends on the type of brackets used to represent the endpoints. It is important to understand the domain of a function because it helps determine the range, identify any restrictions, and ensure proper usage of the function.
Niles

## Homework Statement

Hi

If I have a function defined in one point only given by e.g.

f(x) = 2,

then is the function f defined on a open "interval" or a closed "interval"?

Simple answer: a single point is a closed interval.

Complicated answer: it depends on the topology you put on R.

The function f is defined on a closed interval. This is because the interval includes the single point where the function is defined, making it a closed interval. An open interval would not include the point itself.

## 1. What is the domain of a function?

The domain of a function refers to the set of all possible input values for a given function. In other words, it is the set of values that can be plugged into the function to produce a valid output.

## 2. What is an open interval?

An open interval is a range of values that includes all real numbers between two given endpoints, but does not include the endpoints themselves. It is represented as (a, b), where a and b are the endpoints and are not included in the interval.

## 3. What is a closed interval?

A closed interval is a range of values that includes all real numbers between two given endpoints, including the endpoints themselves. It is represented as [a, b], where a and b are the endpoints and are included in the interval.

## 4. How do you determine if a function has an open or closed interval?

The type of interval for a function depends on the type of brackets used to represent the endpoints. If the endpoints are represented with parentheses, then the interval is open. If the endpoints are represented with square brackets, then the interval is closed.

## 5. Why is it important to understand the domain of a function?

The domain of a function is important because it tells us which values can be used as inputs for the function. It also helps us determine the range of the function, which is the set of all possible output values. Understanding the domain can also help us identify any restrictions or limitations on the function, and ensure that we are using it correctly.

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