Proving Continuity and Linearity of a Function

[ILikePizza]

Homework Statement

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.

Homework Equations

definition of continuity

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!

 

[Office_Shredder]

Not sure for part a off the top of my head, but assuming it's solved...

For part b, start by proving it for c a natural number, then extend it to all rational numbers without using the continuity condition. Use continuity (f(1) is continuous hence by part (a) all of f is) to extend f(xc)=cf(x) for c an irrational number

 

[HallsofIvy]

If f is continuous at x= p, then $lim_{x\rightarrow p}f(x)= f(p)$. Let h= x- p. Then as x goes to p, h goes to 0 and f(x)= f(p+ h)= f(p)+ f(h).
$lim_{x\rightarrow} f(x)= \lim_{h\rightarrow 0}(f(p)+ f(h))= f(p)+ \lim_{h rightarrow 0} f(h)= f(p)$.

What does that tell you about $\lim_{h\rightarrow 0} f(h)$.

Now for any q, look at $\lim_{x\rightarrow q} f(x)$ by letting h= x- q.