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Possible Duplicate:
Functions which are Continuous, but not Bicontinuous

If $f$ is a continuous map from a subset of $\mathbb{R}^n$ to another subset of $\mathbb{R}^n$, must it have a continuous inverse? (in usual topology) Is the same true of metric spaces? When is it true/not true?

Requesting example if not.

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    No: Several examples appear at [Functions which are Continuous, but not Bicontinuous](http://math.stackexchange.com/questions/68800/functions-which-are-continuous-but-not-bicontinuous).2012-02-04
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    In general, it's very false, as others pointed out; but if $f$ is a continuous bijection (a necessary condition for having an inverse) between *open* subsets of $\mathbb{R}^n$, then $f$ does have a continuous inverse: this is known as the invariance of domain theorem (domain being an old name for open subset of $\mathbb{R}^n$), due to Brouwer, and quite hard to prove from first principles.2012-02-04
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    Not only is it false that a continuous map between Euclidean subspaces must have a continuous inverse,this is not even true in general if THE MAP IS A CONTINUOUS BIJECTION BETWEEN THE SPACES!2012-02-04
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    @Mathemagician1234 : You are wrong (and there is no need to shout). A continuous bijection between two manifolds automatically has a continuous inverse. Go read the wikipedia article on the invariance of domain theorem.2012-02-05
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    @AdamSmith: I also want to understand this thing. Let us look at the definition of homeomorphism in [this page](http://en.wikipedia.org/wiki/Homeomorphism#Definition): it says $f$ is a homeomorphism if it is continuous, a bijection, _and_ inverse is also continuous. If the first two implied the third, why did we need the third one in a definition? I guess there are examples which are continuous bijections but inverse isn't continuous.2013-10-11
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    @Swaprava : Sorry, I only noticed your comment now. For arbitrary topological spaces, the first two of your conditions do not imply the third. However, for manifolds they do. This is a fairly nontrivial theorem called "invariance of domain".2013-11-22

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