An elementary example which any grad can think about. Hope it will require 10 mins for most of them to figure out (or recall) the proof:
The set of discontinuous points of a non-decreasing function.
Or, making it more geometric, (under suitable assumptions) the set of the radius $r$ such that a given Borel measure charges the $r$-sphere.
Or, based on the first result, the set of non-differentiable points of a convex function on the real line.