# Cos(x) is not closed function?

• emptyboat
In summary, a function is "closed" if all possible inputs have a corresponding output within its domain. It is important for a function to be closed in order to ensure accurate mathematical operations and a complete understanding of its behavior. A non-closed function can be identified by a non-continuous domain, gaps in its range, or undefined/infinite values. Cosine (cos(x)) is an example of a non-closed function because it has a limited range of values. While non-closed functions can still be useful in certain contexts, closed functions are generally preferred for their accuracy and completeness.
emptyboat
I think cos(x) is closed function in R.
But I heard that cos(x) is not closed function in R.
What do I choose closed set A in R, cos(A) is not closed in R?
Help...

Consider for instance the set $A := \{2n\pi-1/n: n\in\mathbb{N}\}$. It is closed, but cos(A) does not contain its limit point 1.

Thank you, friend. I tried closed intervals...

## 1. What does it mean for a function to be "closed"?

A function is considered "closed" if the set of all its possible outputs is contained within its own domain. In other words, all possible inputs must have a corresponding output in order for a function to be considered closed.

## 2. Why is it important for a function to be closed?

A closed function ensures that every possible input has a defined output, which is necessary for mathematical operations and calculations. It also allows for a more accurate and complete understanding of the behavior of the function.

## 3. How can you tell if a function is not closed?

If a function has a domain that is not continuous or has "gaps" in its range, it is likely not a closed function. Additionally, if there are any undefined or infinite values in the function, it is not considered closed.

## 4. What is the significance of "cos(x) is not closed function" specifically?

Cosine (cos(x)) is a trigonometric function that is defined for all real numbers, but its range is limited to values between -1 and 1. This means that there are inputs for which there is no corresponding output, making it a non-closed function.

## 5. Can a non-closed function still be useful in scientific research?

Yes, non-closed functions can still be useful in certain contexts. For example, they can be used to model real-world phenomena or to approximate more complex functions. However, in most cases, closed functions are preferred as they provide a more complete and accurate representation of the relationship between inputs and outputs.

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