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**HallsofIvy** What is given is that f is a polynomial and that, for all x and y, f(x+ y)= f(x)+ f(y).

What is needed for the proof given here is simply that f is differentiable so "f is a polynomial" is sufficient but not necessary.

Yes, ayushdadhwal, that is a valid proof. You have shown that the derivative is a constant and so f(x)= cx.

It is, by the way, possible to prove this without assuming differentiabllity, only continuity. However, there exist non-continuous functions satisfying f(x+ y)= f(x)+ f(y). They, of course, are nothing like "f(x)= cx".