Exercise 23, § 11

Give an example of a bijective mapping $f$ of a Hausdorff space $X$ onto a Hausdorff space $Y$, such that the image under $f$ of every compact subset of $X$ is a compact subset of $Y$, and the image under $f$ of every connected subset of $X$ is a connected subset of $Y$, but such that $f$ is not continuous (cf. § 9, Exercise 4).