TITLE: Does continuous imply continuous inverse?
QUESTION [7 upvotes]: 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.

REPLY [15 votes]: My favorite example.  Domain is the interval $[0,2\pi[\;$ in $\mathbb R$, range is in $\mathbb R^2$, formula is $f(\theta) = (\cos \theta, \sin\theta).\;$  This is a continuous map of that interval one-to-one onto a circle.  But the inverse is discontinuous.