It is commonly known that the solution for Cauchy's Functional Equation
has solutions . (If you don't know how to prove this, I ask for you to attempt to prove this)
But now, if we instead change this function from being in the rationals to being the reals, we get a strange property.
Let be a function such that
Consider the graph of on the plane. Prove that for any disc, no matter how small, there always exists a point that lies on the graph.
For this proof, you must accept the Axiom of Choice as true.
This question came from a corollary I used in APMO 2016 Q5.