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A function $f : [a,b] \to \mathbb{R}$ is both continuous and $1-1$. How can I prove that it must be monotone?

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    In fact, it is strictly monotone.2012-04-04

3 Answers 3

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Hint. Intermediate Value Theorem. Suppose there exist $c,d,e$, $a\leq c\lt d \lt e\leq b$ such that $f(c)\lt f(d)$ and $f(d)\gt f(e)$. Must there be a point between $c$ and $d$ where it takes the same value as at a point between $d$ and $e$? What if $f(c)\gt f(d)$ and $f(d)\lt f(e)$?

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By supposing that it isn't, and deriving a contradiction, using what you know about continuous one-one functions.

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Hint If $f:[a,b]\to\mathbb{R}$ is continuous, then $\text{Im}(f(x))$ is closed interval.