let f be a differentiable and continuous function on closed bounded interval [a,

Question

let f be a differentiable and continuous function on closed bounded interval [a,b]. use topology (mean value theorem, e.g) to show that if the derivative f' never equals 0 then f is a monotonic function.

Explanation / Answer

Most math teachers will prefer a direct proof over a proof by contradiction. This is an excellent question because both approaches give insight to the problem. When I teach analysis, I usually do both to reinforce the definition of continuity. If f were not bounded, then for each positive integer N, there is x_N in [a,b] satisfying |f(x_N)|>N. The set {x_N} is an infinite set in [a,b] so by Bolzano-Weierstrass, It has an accumulation point c that is contained in [a,b]. Let {x_m} be a subsequence of {x_N} converging to c. Since lim_{m o infty} x_m does not exist, lim_{x o c} f(x) does not exist, so f(x) is not continuous at c, which is a contradiction. Your first real analysis course may be the most difficult course you take as an undergraduate. Good luck! Challenge: Now prove that f is uniformly continuous on [a,b]. (Note that this has previously been answered in Answers and the proof will be in most analysis texts, but you should try it on your own first! :) )

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