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Thread: Differentiable Proof 2

  1. #1
    Member thaopanda's Avatar
    Sep 2009
    Worcester, Massachusetts

    Differentiable Proof 2

    Suppose that f: R \rightarrow R is differentiable and that for every a,b \in R there holds f (a + b) = f(a) + f(b).

    Prove that f '(x) = f '(0) for all x \in R. What kind of function is f?
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  2. #2
    Nov 2009
    Start by noticing that f(0)=0. Indeed, putting a=b=0, one gets f(0)=2f(0), hence f(0)=0.
    Now put a=x, b=h:

    Divide by h to get

    Take the limit h->0 to finally get

    This means that f'(x)=a is a constant: f(x)=cx+d, but d must be zero in order to the first relation to hold.

    Hence f(x)=cx. Those are called linear functions.
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