MathDB

Problem 5

Part of 2009 CIIM

Problems(1)

CIIM 2009 Problem 5

Source:

6/9/2016
Let f:R→Rf:\mathbb{R} \to \mathbb{R}, such that
i) For all a∈Ra \in \mathbb{R} and all ϵ>0\epsilon > 0, exists δ>0\delta > 0 such that ∣x−a∣<δ⇒f(x)<f(a)+ϵ.|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.
ii) For all b∈Rb\in \mathbb{R} and all ϵ>0\epsilon > 0, exists x,y∈Rx,y \in \mathbb{R} with b−ϵ<x<b<y<b+ϵ b - \epsilon < x < b < y < b + \epsilon, such that ∣f(x)−f(b)∣<ϵ|f(x)-f(b)|< \epsilon and ∣f(y)−f(b)∣<ϵ.|f(y)-f(b)| < \epsilon.
Prove that if f(a)<d<f(d)f(a) < d < f(d) there exists cc with a<c<ba < c < b or b<c<ab < c < a such that f(c)=df(c) = d.
CIIM 2009CIIMUndegraduate