- #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.