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Let $f:[0,1]\to \mathbb{R}$ be an injective function prove that either $f(0)\le f(x)\le f(1)$ or $f(1)\le f(x)\le f(0)$ for all $x\in [0,1]$

how to start with this problem I really have no idea

learner
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1 Answers1

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Counterexample: $$f(x) = \begin{cases} -x &, x \in [0,0.5) \\ 2-x &, x\in [0.5,1]\end{cases}$$

Siong Thye Goh
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