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

Let $S=\{0\}\cup \{1/n:n\ge 1\}$ 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.

Using the preceding $f$, define $g(0)=f(0)+1$ and $g(x)=f(x)$ for $x>0$. Changing the value only at zero makes zero discontinuous while preserving continuity at every $1/n$ and discontinuity everywhere else. Thus the continuity set is precisely $\{1/n:n\ge 1\}$.