$f \circ g = g \circ f$ $\iff$ $f$ and $g$ are two linear functions?

Question

Is $$f(x + y) = f(x) + f(y), g(x + y) = g(x) + g(y) \Leftrightarrow \ g\;o\;f = f\;o\;g $$ true?

I have a feeling that it must be true, at least the forward way, but I couldn't come up with a proof..

Any ideas?