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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.

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    $\begingroup$ The Axiom of Choice (AC) (equivalently Zorn's Lemma, which is really an axiom) declares the existence of certain kinds of sets. And you need AC to prove such functions exist. Search this site and you will find proofs. Lebesgue measure is not involved. But an example cannot be given by elementary formulas because the negation of AC is also consistent with the other axioms of Set Theory. $\endgroup$
    – DanielWainfleet
    Apr 5, 2018 at 17:59
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    $\begingroup$ Such a function is everywhere discontinuous and is unbounded on every interval of non-zero length. $\endgroup$
    – DanielWainfleet
    Apr 5, 2018 at 18:02

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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.)”

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