Continuous bijection whose inverse is not continuous at uncountably many points

Question

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible that $f^{-1}$ fails to be continuous at uncountably many points of $Y$?

Apologies if the question is trivial!

Answer 1

Let $X$ be the reals with the discrete topology and $Y$ the reals with the usual topology, and let $f$ be the identity map; then $X$ and $Y$ are metrizable, $f$ is a continuous bijection, and $f^{-1}$ is nowhere continuous.

Answer 2

Let $X$ be $[0,1)$ and Y $s^1$ , f send each point on to interval to 1 sphere f is bijective and countinuous and $f^{-1}$ is'nt countinuous. By taking the uncounteble product you will get a countinious bijection witch its inverses isnt countinous at uncountable number of points.