Is it true or not that open map $f:\mathbb{R} \rightarrow \mathbb{R}$ is a one one map? If it is one one explain how it is one one.
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Well the image is connected and therefore an interval. If $f(a)=f(b)$ then then think about what happens to the interval $(a,b)$, the image must have a finite sup or inf. – Gregory Grant May 02 '15 at 14:29
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There are functions whose range on any nonempty open interval is all of $\Bbb R$. See Gelbaum and Olmsted, Counterexamples in Analysis, example 27, Chapter 8 pg. 104. (This may be overkill.) – David Mitra May 02 '15 at 14:32
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its very long example.. – neelkanth May 02 '15 at 14:47
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please explain.... – neelkanth May 02 '15 at 14:54
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but if we define any value at zero it will not be open.. – neelkanth May 02 '15 at 15:12