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Is there any name of a function who is continuous and its inverse function is also continuous?

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a continuous bijection with continuous inverse is called a homeomorphism – yoyo Mar 30 '11 at 2:10
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That's called a homeomorphism.

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Yes. Such a function is a homeomorphism. See Wikipedia

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It is called a bicontinuous function.

Just look at the first sentence in the wikipedia entry for continuous function.

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Yes and what happens if the function is not many-to-one? – Thales Mar 30 '11 at 6:21
@Thales I'm sorry but I don't understand your comment. – Adrián Barquero Mar 30 '11 at 6:47

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