Let $f$ be a continuous, invertible map on $[0,1)$. How do we prove that $f$ must be increasing, and thus $f(0)=0$ and $\lim_{x \to 1}f(x)=1$? It's simple to see that $f$ must be either strictly increasing or decreasing. I understand how this must work, having difficulty with formality.
Continuous invertible map on [0,1) must be increasing?
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real-analysis
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0If $f(x)=0$ and $x\neq 0$, what happens on $[0,x)$ and $(x,1)$? – 2017-02-12
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0Well there must be a pretty image of 0. If f (x) = 0 and 0
f (x) and f (z) > f (x) which by intermediate value theorem will show f is not invertable. – 2017-02-12