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Yes, f can be any function at all here, and indeed this allows for the sort of more complicated things you mention, and indeed this is where a lot of the difficulty lies. Duffin and Schaeffer, for whom the conjecture is named, had already proved it subject to a restriction on f that kinda-sorta requires it not to vary too wildly.

(One other bit of precision that I glossed over: the conjecture is actually about approximations p/q for which p and q have no common factor and that genuinely makes a difference here.)

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