Basic question regarding continuous inverses

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A function is defined as a homeomorphism if it is continuous, bijective, and possesses a continuous inverse. In the case of a mapping f: R -> (0,1), the continuity of the inverse function is required only on the codomain (0,1). Therefore, it is unnecessary to impose continuity on any larger subset beyond (0,1), as the inverse function is only defined within this interval.

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TaylorWatts
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Regarding the definition of homemorphism, when we say a function is a homeomorphism if it is continuous, bijective, and has a continuous inverse I assume that means over the codomain only.

For example if we have a map from f: R -> (0,1) does f inverse need to be continuous on (0,1) only?
 
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Well f inverse, if it exists, is only defined on (0,1). So it wouldn't make sense to require it to be continuous on a larger subset that (0,1).
 

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