Let f be a function with domain the reals and range the reals.Assume that f has a local minimum at each point x in its domain.(This means that, for each x ? R, there is an E = Ex > 0 suchthat, whenever | x?t |< E then f(x) ? f(t).) Do not assume thatf is differentiable, or continuous, or anything nice like that.Prove that the image of f is countable. (Hint: When I solved thisproblem as a student my solution was ten pages long; however, thereis a one-line solution due to Michael Spivak.)
