Sunday, January 17, 2016

functional equations - Do there exist functions satisfying f(x+y)=f(x)+f(y) that aren't linear?



Do there exist functions f:RR such that f(x+y)=f(x)+f(y), but which aren't linear? I bet you they exist, but I can't think of any examples.



Furthermore, what hypotheses do we need to put on f before no such functions exist? I feel continuity should be enough.


Answer



Yes continuity is enough: You can quickly show that f(x)=xf(1) for xN, then for xZ and then for xQ; assuming continuity, this implies validity for all xR.




Any other functions only exist per Axiom of Choice: View R as a vector space over Q and take any Q-linear map (which need not be R-linear).


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