(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given that f(x + y) = f(x) + f(y), prove that

(a) if this function is continuous at some point p, then it is continuous everywhere

(b) this function is linear if f(1) is continuous.

2. Relevant equations

definition of continuity

3. The attempt at a solution

(a) I think that contradcition(sp?) would work nicely here. But i'm not sure exactly how it would work. I mean, there exists a point q such that there exists a x > 0 such that for all d > 0, .... what would go in the "..."? |f(d) - f(q)| < x?

Beyond that, where do I go from there . any ideas? Is contradiction the right way to go?

(b) The only way i can think of making this work is showing if f(xc) =c f(x), we win. But again, how would you show this?

thanks in advance!

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# Homework Help: Linear function

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