- #1

jdinatale

- 155

- 0

## Homework Statement

Determine all continuous functions g: R -> R such that g(x + y) = g(x) + g(y) for all [itex]x, y \in \mathbf{R}[/itex]

## The Attempt at a Solution

g(x) = g(x + 0) = g(x) + g(0). Hence G(0) = 0.

G(0) = g(x + -x) = g(x) + g(-x) = 0. Therefore g(x) = -g(-x).

It seems obvious that the only solutions that satisfy these properties are in the form of [itex]g(x) = \alpha x[/itex] for some [itex]\alpha \in \mathbf{R}[/itex].

My issue is determining that these are the ONLY such functions. I have to somehow rule out every other possible function.

I can rule out all functions in the form of g(s) = ax + b for [itex]b \not= 0[/itex] since solutions in that form would imply that

g(s + t) = a(s + t) + b = as + at + b

and

g(s + t) = g(s) + g(t) = as + b + at + b = as + at + 2b

which is impossible. But I have to somehow rule out the infinitely many other types of possible functions.