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Calculus Proof

  1. Oct 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose that |f (x) - f (y)|  (x-y)^2 for all real numbers x and y: Prove
    that f is a constant function.

    2. Relevant equations

    No relevant equations..


    3. The attempt at a solution

    I'm really stuck..i'm thinking you're suppose to use mathematical induction.?
     
  2. jcsd
  3. Oct 19, 2009 #2

    Dick

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    Can you show that |f(x)-f(y)|<=(x-y)^2 for all x and y implies that f is differentiable, and that it's derivative equals zero everywhere?
     
  4. Oct 19, 2009 #3

    lurflurf

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    Dick's approach is popular, but I never liked it. I do not think this problem merits the use of the derivative. I like to show inductively that
    |f(x)-f(y)|<=(x-y)^2 implies
    |f(x)-f(y)|<=(x-y)^2/2^n for any natural number n from which the result is obvious.
    hint
    |f(x)-f(y)|=|[f(x)-f((x+y)/2]+[f((x+y)/2)-f(y)]|<=|f(x)-f((x+y)/2|+|f((x+y)/2)-f(y)|
     
  5. Oct 19, 2009 #4

    Dick

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    That is a nice alternative approach. There's more than one way to skin a cat.
     
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