The additive function is bounded

In summary, the conversation discusses how to prove that an additive function with a limit at each real number must have a number a greater than zero and M greater than zero, such that |f(x)| is less than or equal to M for all x in the interval [-a,a]. The speaker also asks if it can be shown that for any rational number r, f(rx) is equal to rf(x). From this, it can be inferred that if the limit exists at any x, then the function is continuous at that point and for all x. This leads to the conclusion that f(x) is equal to c times f(x), where c is a real constant.
  • #1
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Hi,
If I have an additive function which is [itex]f(x+y)=f(x)+f(y)[/itex],
the question is
how can we prove that if this function has a limit at each real number then there is a number a greater than zero and M greater than zero
such that
[itex]|f(x)|\leq M[/itex], for all [itex]x\in[-a,a][/itex],
 
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  • #2
Can you show that, for any rational number, r, f(rx)= rf(x)? From that, you should be able to show that if the limit exists at any x, then f is continuous there and so is continuous for all x. From that it follows that f(x)= cf(x) where c is a real constant.
 

1. What does it mean for the additive function to be bounded?

The additive function being bounded means that there exists a constant value that limits the function and prevents it from increasing without bound. In other words, the function has a finite range and does not grow infinitely large.

2. What are the implications of the additive function being bounded?

If the additive function is bounded, it means that it has a fixed limit and cannot exceed that limit. This can be useful in mathematical calculations and proofs, as it allows for more precise and accurate results.

3. How can you prove that the additive function is bounded?

One way to prove that the additive function is bounded is by using the definition of a bounded function, which states that for any value of x, there exists a constant K such that the absolute value of the function is less than or equal to K. If this condition holds true for the additive function, then it is indeed bounded.

4. Can the additive function be both bounded and unbounded?

No, the additive function can only be either bounded or unbounded. It cannot be both at the same time. If the function has a finite range and a fixed limit, it is bounded. If the function does not have a limit and can increase without bound, it is unbounded.

5. How does the boundedness of the additive function affect its graph?

If the additive function is bounded, its graph will have a finite range and be limited to a specific area on the coordinate plane. The graph will also have a horizontal asymptote, which is a line that the graph approaches but never touches. This asymptote represents the bound or limit of the function.

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