Do there exist functions f:R→R 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)=x⋅f(1) for x∈N, then for x∈Z and then for x∈Q; assuming continuity, this implies validity for all x∈R.
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).
No comments:
Post a Comment