Let $S=\{0\}\cup\{1/n:n\ge1\}$ and let $D$ be the Dirichlet <function>, equal to $1$ on rationals and $0$ on irrationals. Then $f(x)=d(x,S)D(x)$ is continuous exactly on $S$: the distance factor squeezes it to zero on $S$, while away from $S$ it is a positive continuous factor times an everywhere-discontinuous <function>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 10E / e / i / Solution

 Solution (source code)