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Suppose $I$ and $J$ are intervals in $\mathbb{R}$ and $f: I \rightarrow J$ is a continuous bijection.

Can we state that $f$ is a homeomorphism?

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Yes. To prove it you should show that $f$ is (strictly) monotone. Then the inverse is also monotone and hence could only possibly have jump discontinuities. But a jump discontinuity cannot occur, otherwise $f$ would be constant on an interval and hence not injective.

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I understand. The function $f$ needs to be constant on an interval due to its continuity. Thank you. So in fact this would be an example of when the additional requirement of being an open or closed map can be dropped. – user62313 Feb 14 '13 at 16:41

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