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Proof using continuous f(x+y)=f(x)+f(y)

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Let f be a continuous function lR (all real numbers) --> lR such that f(x+y) = f(x) + f (y) for x, y in lR.
    prove that f(n) = n*f(1) for all n in lN (all natural numbers)

    2. Relevant equations

    f is continuous

    also note and prove that f(0) = 0

    3. The attempt at a solution

    I figured out the general proof using induction, assuming that the base case f(0) is true:

    Assuming f(n) = nf(1), prove that f(n+1) = (n+1)f(1):

    we know f(x+y) =f(x) + f (y)
    so f(n+1) =f(n) + f (1)
    and according to inductive hypothesis, f(n) = nf(1)
    s0 f(n) + f (1)
    = nf(1) + f (1)

    But I still don't understand why the base case f(0) = 0 is true..?
    Last edited: Apr 26, 2010
  2. jcsd
  3. Apr 26, 2010 #2


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    Homework Helper

    f(0) = f(0 + 0) = f(0) + f(0) => f(0) = 0.
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