1. Nov 30, 2004

### Ed Quanta

So if f is a monotone function which takes elements of the Reals to the Reals. If f is additive, how do I show that f is continuous?

2. Nov 30, 2004

### Hurkyl

Staff Emeritus
Probably by assuming it's not continuous. You know what f(0) is, right? That's often a key place to analyze. Making a conjecture for a formula for f might help too.

Last edited: Nov 30, 2004
3. Nov 30, 2004

### Ed Quanta

Nope, I don't know what f(0) is.

4. Nov 30, 2004

### Hurkyl

Staff Emeritus
You can figure it out from the definition of additive, though!

You can probably figure out a lot of values of f in terms of f(1), too!

5. Nov 30, 2004

### Ed Quanta

I see that f(0)=0 since f(0)=f(0)+f(0) when x=y=0

And I also see that f(2)=2f(1) and f(3)=3f(1) and f(n)=nf(n) but this is only true when n is an integer. And n is not strictly limited to being an integer, it can be any rational number I believe.

And I am not sure what to do with this information.

6. Dec 1, 2004

### Hurkyl

Staff Emeritus
So, you know that for n rational, f(n) = n f(1).
You also know that f is monotone -- does that help you with figuring out values of f at irrational numbers?

7. Dec 1, 2004

### Ed Quanta

Perhaps it would help someone smarter, but not me. I really do appreciate your help very much. I hope I am not annoying you.

1)I can't find a logical connection between the rationals and irrationals here. I am not even convinced that f(n)=nf(1) for all rationals. How do I know f(1/2)=1/2f(1) for instance?

2)I am not sure how to use the fact that the function is monotone to help me with this proof. I know what monotone means, but how do I know that no jump discontinuities exist? I know that a montone function on the Reals can only have countably many jump discontinuities, and the irrationals aren't countable so f must be continuous at some irrational number. How do I jump to the conclusion that f is continuous for all of them?

Last edited: Dec 1, 2004
8. Dec 1, 2004

### arildno

Here's how to show that f(a)=af(1) for a rational:
Let p,n be integers:
$$pf(1)=f(p)=f(\frac{p}{n}+++\frac{p}{n})=nf(\frac{p}{n})$$

The "+++" means we have n terms in our argument.

9. Dec 1, 2004

### Hurkyl

Staff Emeritus
Suppose one does. What does that say about the values at the rational points?

or

How can you identify irrational points using only rational points?