Cos(x) is not closed function?

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.
  • #1
emptyboat
28
1
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...
 
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  • #2
Consider for instance the set [itex]A := \{2n\pi-1/n: n\in\mathbb{N}\}[/itex]. It is closed, but cos(A) does not contain its limit point 1.
 
  • #3
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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