Cauchy's functional equation, f (x + y) = f (x) + f (y), f : R → R looks very simple, and it has a class of simple solutions, f (x) = λx, but there are many other and more interesting solutions. In these notes, I will show you what some of these 'wild' solutions look like, and I will use them to prove that there exist a set A ⊂ R, such that neither A nor R \ A contains a measurable subset with positive measure. Section 1 is about Cauchy's functional equation on the rational numbers, in section 2 I show that there some wild solutions on R, and in section 3 I will show that their graphs are dense in R 2. In section 4 I'll show that these functions are ugly from a measure theoretical point of view, and in section 5, I'll show that some of these functions are wilder than others. E.g., I will prove that there is a solution to Cauchy's functional equation, that intersects any continuous function from R to R.