# Cauchy's functional equation real to real

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.

• 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. – DanielWainfleet Apr 5 '18 at 17:59
• Such a function is everywhere discontinuous and is unbounded on every interval of non-zero length. – DanielWainfleet Apr 5 '18 at 18:02