1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    Edit:
    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)
    =(n+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

    radou

    User Avatar
    Homework Helper

    f(0) = f(0 + 0) = f(0) + f(0) => f(0) = 0.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook