Cauchy's functional equation: $$f(x+y)=f(x)+f(y)$$ On wikipedia (and some other websites) it says that there are non-linear solutions for real to real. But I don't quite understand about additive functions and Lebesgue measure.Can someone give me an example of a non-linear solution and explain the set of non-linear solutions throughly?
Thank you in advance.
No, nobody can give an example, if, when you write “example”, what you mean is a functions that you can work with. In fact, there are variants of set theory for which no such a function exists. Or, as the Wikipedia article says, after proving the existence of such functions: “Note, however, that this method is nonconstructive, relying as it does on the existence of a (Hamel) basis for any vector space, a statement proved using Zorn's lemma. (In fact, the existence of a basis for every vector space is logically equivalent to the axiom of choice.)”