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Apr 14, 2023 at 14:54 comment added Iosif Pinelis @dnkywin : You should not change the question so as to invalidate a valid answer.
Apr 14, 2023 at 6:46 comment added dnkywin Thanks for your answer! I realized that I actually wanted condition (1) to be $g\ge c\cdot f_S$ for some finite $S$ and some $c > 0$. Any continuous $g$ that is a product of local functions would satisfy this property, so a counterexample would have to be more involved. I’ve edited the original question to reflect this.
Apr 14, 2023 at 6:05 history answered Christian Remling CC BY-SA 4.0