# Open/closed set

1. Nov 8, 2011

### Ted123

1. The problem statement, all variables and given/known data

If $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ are continuous functions, give examples to show that the set $\{ (f(x),g(x)) : x\in\mathbb{R} \}$ might or might not be closed in $\mathbb{R}^2$.

3. The attempt at a solution

Letting $f(x)=g(x)=0$ gives the set equal to $\{ (0,0) \}$, a singleton and singleton sets are closed.

What functions would make the set open?

2. Nov 8, 2011

### Dick

That's a pretty trivial example. You should probably think about it a little more before someone just gives you the answer. Suppose f(x) and g(x) have a horizontal asymptote as x->inf. There's other ways it can fail to be closed as well, but that should get you started.

3. Nov 9, 2011

### Ted123

Well if the functions have horizontal asymptotes they won't be continuous on the real line for a start.

If we take f(x) a constant function, what condition must g(x) satisfy in order that the set isn't closed?

4. Nov 9, 2011

### micromass

Staff Emeritus
One VERY IMPORTANT THING:

Open is NOT the same as "not closed".

You don't need to find a set that is open, you need to find a set that is not closed. These are completely different questions!!!!

5. Nov 9, 2011

### Dick

I said HORIZONTAL asymptote. Why do you think that means it wouldn't be continuous?

6. Nov 9, 2011

### Ted123

Sorry, of course it can be continuous.

Is the set $\{ (0,x):x\in\mathbb{R} \}$ not closed?

7. Nov 9, 2011

### Dick

It doesn't look closed to me. What do you think? Keep micromass's comment in mind too. A set that is 'not closed' doesn't have to be open.

8. Nov 9, 2011

### Dick

I changed my mind. I thought you were trying to write an open subsegment of a straight line. You're aren't, are you? Your set is the x-axis. It's closed in R^2. How would you show that?

9. Nov 9, 2011

### Ted123

I've just realised that I'm confusing 'open' and 'not closed'.

With $f$ a constant function, I need to find $g$ such that $g(\mathbb{R})$ is not closed. What is an example a function from R to R that isn't closed?

10. Nov 9, 2011

### Dick

So you aren't going to try the horizontal asymptote suggestion???

11. Nov 9, 2011

### Ted123

$g(x)=e^x$.

Then $\{(0,e^x) :x\in\mathbb{R} \}$ is not closed is it?

12. Nov 9, 2011

### Dick

No, it isn't. Why isn't it closed?

13. Nov 9, 2011

### Ted123

The boundary of $A = \{(0,e^x) : x\in \mathbb{R} \}$ is $\partial A =\{(0,0)\}$ and $(0,0) \notin A$ so $A$ doesn't contain all its boundary and is therefore closed.

14. Nov 9, 2011

### Dick

Good answer! Except that the boundary of A is actually all of A AND (0,0). Every point in A is also a boundary point of A. But the critical thing is that (0,0) is on the boundary of A but not in A.

Last edited: Nov 9, 2011
15. Nov 9, 2011

### Ted123

Would you express the boundary as $\partial A = A\cap \{(0,0)\}$?

16. Nov 9, 2011

### Dick

Certainly not, it's a union not an intersection.