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Take a positive function $g$ on the prime powers that tends to infinity, set $f(q)=g(q)\log q$ for every prime power $q$, and extend $f$ additively. The hypothesis $\limsup_{p,k}f(p^k)/\log p^k=\infty$ holds by construction. When $g$ grows slowly enough, the values of $f$ at consecutive integers differ by $o(\log n)$, so $\limsup_n(f(n+1)-f(n))/\log n=0$, as the site's remark says; Wirsing describes the counterexample as one with $\Delta f=o(\log)$. The ratio $f(n+1)/f(n)$ stays bounded as well, so both limits superior the problem asks about are finite and both questions have answer no. Wirsing records this construction on pp. 235--236 of Additive and completely additive functions with restricted growth, in Recent progress in analytic number theory, Vol. 2 (Durham, 1979), Academic Press (1981), 231--280, and credits the observation to Erdős; the linked copy is the Internet Archive's scan of the volume. The volume gives the year 1981 and no day; the page's date carries the first day of that year.

Acceptance. Reviewed: the site's curator, Thomas F. Bloom, states on the discussion thread (2025-12-27) that the problem was solved by Wirsing in 1981, and the problem page records the answer no to both questions with this construction (last edited 2026-04-01). The 2025 independent rediscovery of the construction and its Lean formalization are recorded on Archivara 2025. The problem page is Problem 897.