(adsbygoogle = window.adsbygoogle || []).push({}); 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..?

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# Homework Help: Proof using continuous f(x+y)=f(x)+f(y)

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