Inverse of uniformly continuous function is uniformly continuous?
Assume that $ X,Y$ are metric spaces and let $f:X\to Y$ such that $f$ is bijective and uniformly continuous. Then can we predict that $f^{-1}$ is uniformly continuous? Suppose not, what condition I should add to make $f^{-1}$ is uniformly continuous.
No. Consider $f:\Bbb R\to(-1,1)$ with $$f(x)=\frac1\pi\tan^{-1}(x)$$