Basic question regarding continuous inverses

[TaylorWatts]

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?

 

[quasar987]

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).