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If $f:[a,b]\rightarrow R$ satisfies the intermediate-value properly on $[a,b]$ and $f$ is injective on $[a,b]$, then show that f is strictly monotone on $[a,b]$
I somehow cannot invoke the injective property. If anyone can help, I'll be glad. Thanks in advance

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Assume that $f$ is not strictly monotone on $[a,b]$. Then $f$ is also not monotone on $[a,b]$, since no two of its function values are equal. Thus without loss of generality there are points $x_0\lt x_1\lt x_2$ with $f(x_0)\lt f(x_1)\gt f(x_2)$. What does this tell you about how often $f$ takes the values between $f(x_1)$ and $\max(f(x_0),f(x_2))$?