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  3. let f be a function with domain the reals and rang

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

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Advance Math

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.)

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