Let f be a function with domain the reals and range the reals. Assume that f...

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 such that, whenever | x−t |< E then f(x) ≤ f(t).) Do not assume that f is differentiable, or continuous, or anything nice like that. Prove that the image of f is countable. (Hint: When I solved this problem as a student my solution was ten pages long; however, there is a one-line solution due to Michael Spivak.)

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Colby Messinger answered 1 year ago

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