Question

(Exercise 5.8 of the text) If f is a real-valued function on (a, b) and c E (a, b), we say that f is strictly increasing at c if there exists δ 〉 0 such that if c-5 〈 x 〈 c, then f(x) 〈 f(c) and if c 〈 x 〈 c+6, then f(c) 〈 f(x). We say that f is strictly increasing on (a, b) if whenever x, y E (a, b) with x 〈 y, we have f(x) 〈 f(y) Prove that if f is strictly increasing at each point of (a, b), then f is strictly increasing on (a, b) Hint: Argue by contradiction. Then there exist c, d (a, b) with c 〈 d and f(c) f(d). Now consider Wl lub {xla 〈 x < d and f(x) f(d)}.]

Answer 1

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