This is problem 9 from Shilov's Elementary Real and Complex Analysis. The Hints given seem a bit off the wall and I am curious to see what the community comes up with sans those suggestions.  [ f(x) is monotonic on (-inf, inf) and satisfies the functional property\  f(x +y) = f(x) + f(y). Show that f(x) = ax for a constant, real a.]
Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real.  We know $f(0)=0$, and if one assumes differentiability it is quite easy.      But how does monotone make up for differentiability?
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