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f(x+y) = f(x) + f(y) + f(kxy) , differentiable

Source: 1963 Swedish Mathematical Competition p4

March 21, 2021
functional equationfunctionalalgebraanalysis

Problem Statement

Given the real number kk, find all differentiable real-valued functions f(x)f(x) defined on the reals such that f(x+y)=f(x)+f(y)+f(kxy)f(x+y) = f(x) + f(y) + f(kxy) for all x,yx, y.
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