# Thread: Bounded, Continuous and Closed Functions

1. ## Bounded, Continuous and Closed Functions

I need a clearer definition of a bounded, continuous and closed function. The lecture Wendnsday wasnt real clear.

What I know, a function is bounded if it lives between two horizontal lines. An example would be the function f(x)=sin(x). Its bounded between 1 and -1.
I also know that its continuous, but why is it closed? What does that mean?

The function f(x)=1/x D=(0,1) is bounded and continuous on its Domain, is it closed?

A more clear explination would be greatly appreciated.

2. A closed set contains all of its limit points. An open set does not. Basically, closed intervals are closed, and open intervals are open. So $(0,1)$ is open. $\sin x$ is closed, because it contains its boundary. See this

A closed and bounded set is said to be compact.

3. Originally Posted by Ranger SVO
I need a clearer definition of a bounded, continuous and closed function. The lecture Wendnsday wasnt real clear.

What I know, a function is bounded if it lives between two horizontal lines. An example would be the function f(x)=sin(x). Its bounded between 1 and -1.
I also know that its continuous, but why is it closed? What does that mean?

The function f(x)=1/x D=(0,1) is bounded and continuous on its Domain, is it closed?

A more clear explination would be greatly appreciated.
A "boundary point" is like an endpoint so $[1,2]$ has two boundary points 1 and 2. Also $(1,2)$ has two 1 and 2 boundary points. An interval (or a set) which contains its boundary points is said to be "closed" so $[1,2]$ is closed. While $(1,2)$ is not closed. An interval (or a set) which contains none of its bounday points is "open" so $(1,2)$ is open. Note, $[1,2)$ is neither open nor closed.

4. Thats the explination that I needed, thank you