Additive Montone Functions Proof

[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?

 

[Hurkyl]

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.

 

[Ed Quanta]

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

 

[Hurkyl]

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!

 

[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.

 

[Hurkyl]

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?

 

[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?

 

[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.

 

[Hurkyl]

how do I know that no jump discontinuities exist

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?