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Math Help - The Konstant Function

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    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.
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    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    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?
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    Quote Originally Posted by ecMathGeek View Post
    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.
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    Super Member Rebesques's Avatar
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    f(x)=x^2 satisfies this and is not Konstant.
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    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    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.

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

    |f(x)-f(y)|<=(x-y)^2
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    Yes that would make a difference. See the file.
    Attached Files Attached Files
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