1. ## The Konstant Function

In my Real Analysis book I found a very interesting problem.

Let f be a function on R such that:
|f(x-y)| <= (x-y)^2 for all x,y in R.

Show that, f is a constant function on R.

2. Originally Posted by ThePerfectHacker
In my Real Analysis book I found a very interesting problem.

Let f be a function on R such that:
|f(x-y)| <= (x-y)^2 for all x,y in R.

Show that, f is a constant function on R.
Is f a function of one variable? For example, might it be something like f(a), where in this case we are letting a = x - y?

3. Originally Posted by ecMathGeek
Is f a function of one variable? For example, might it be something like f(a), where in this case we are letting a = x - y?
Yes. If it were a function in two variables I would have said a function on R^2.

4. f(x)=x^2 satisfies this and is not Konstant.

5. Originally Posted by ThePerfectHacker
Yes. If it were a function in two variables I would have said a function on R^2.
I figured that was the case, but I wanted to be sure there was no typo in the problem.

Originally Posted by Rebesques
f(x)=x^2 satisfies this and is not Konstant.
I was thinking the same thing. But I think we are misunderstanding the logic of the problem.

6. Originally Posted by Rebesques
f(x)=x^2 satisfies this and is not Konstant.
Sorry I meant to write:

|f(x)-f(y)|<=(x-y)^2

7. Yes that would make a difference. See the file.