Additive Montone Functions Proof

In summary, the conversation discusses how to show that an additive, monotone function is continuous. The key steps are to use the definition of additive to find values of the function at rational points, and then use the fact that the function is monotone to show that no jump discontinuities exist at irrational points. This can be done by considering the values of the function at rational points and using them to identify potential jump discontinuities at irrational points.
  • #1
Ed Quanta
297
0
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?
 
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  • #2
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.
 
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  • #3
Nope, I don't know what f(0) is.
 
  • #4
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
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
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
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?
 
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  • #8
Here's how to show that f(a)=af(1) for a rational:
Let p,n be integers:
[tex]pf(1)=f(p)=f(\frac{p}{n}+++\frac{p}{n})=nf(\frac{p}{n})[/tex]

The "+++" means we have n terms in our argument.
 
  • #9
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?
 

1. What is an additive monotone function?

An additive monotone function is a function that satisfies two properties: additivity and monotonicity. Additivity means that the function preserves addition, meaning that f(x+y) = f(x) + f(y) for all values of x and y. Monotonicity means that the function preserves order, meaning that if x is less than or equal to y, then f(x) is less than or equal to f(y).

2. What is the importance of studying additive monotone functions?

Additive monotone functions are important in many areas of mathematics and economics. They can be used to model and analyze various real-world phenomena, such as population growth, interest rates, and stock prices. Understanding the properties and behavior of these functions can help us make predictions and informed decisions in these fields.

3. How do you prove that a function is additive and monotone?

To prove that a function is additive and monotone, you must show that it satisfies the two properties mentioned earlier: additivity and monotonicity. This can be done through mathematical induction or by using the definition of additivity and monotonicity to show that the function holds for all values of x and y.

4. Can an additive monotone function be differentiable?

Yes, an additive monotone function can be differentiable. In fact, all differentiable functions are also additive and monotone. However, not all additive monotone functions are differentiable, as there are some functions that satisfy the properties of additivity and monotonicity but are not continuous, and therefore not differentiable.

5. Are there any real-life applications of additive monotone functions?

Yes, there are many real-life applications of additive monotone functions. In economics, they are used to model and analyze growth and decline in various industries. In finance, they are used to calculate interest rates and investment returns. In population studies, they are used to analyze population growth and migration patterns. Overall, understanding the properties and behavior of these functions can provide valuable insights in many fields.

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