Cutting sets of continuous functions on the unit interval
Marek Balcerzak, Piotr Nowakowski, Micha{\l} Pop{\l}awski

TL;DR
This paper characterizes the sets where continuous functions cross zero, showing that any closed nowhere dense set with certain accumulation properties can be realized as the zero-crossing set of a smooth function.
Contribution
It provides necessary and sufficient conditions for a set to be the zero-crossing set of a continuous or smooth function, generalizing previous work and constructing functions with prescribed zero-crossing sets.
Findings
E(f) is a closed nowhere dense subset of the zero set of f.
Any closed nowhere dense set with accumulation points at 0 and 1 can be realized as E(f) for some smooth f.
The paper extends previous results by characterizing zero-crossing sets for continuous and smooth functions.
Abstract
For a function , we consider the set of points at which cuts the real axis. Given and a Cantor set with , we obtain conditions equivalent to the conjunction (or ) and . This generalizes some ideas of Zabeti. We observe that, if is continuous, then is a closed nowhere dense subset of where each is an accumulation point of . Our main result states that, for a closed nowhere dense set with each being an accumulation point of , there exists such that .
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