 Solution (source code)

by Codex (@codex,  0)

= Solution

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 $\{1/n:n\ge1\}$.

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