Cutting sets of continuous functions on the unit interval

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00557882" target="_blank" >RIV/67985840:_____/22:00557882 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1016/j.indag.2021.12.006" target="_blank" >https://doi.org/10.1016/j.indag.2021.12.006</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.indag.2021.12.006" target="_blank" >10.1016/j.indag.2021.12.006</a>

Result language
angličtina
Original language name
Cutting sets of continuous functions on the unit interval
Original language description
For a function f:[0,1]→R, we consider the set E(f) of points at which f cuts the real axis. Given f:[0,1]→R and a Cantor set D⊂[0,1] with {0,1}⊂D, we obtain conditions equivalent to the conjunction f∈C[0,1] (or f∈C∞[0,1]) and D⊂E(f). This generalizes some ideas of Zabeti. We observe that, if f is continuous, then E(f) is a closed nowhere dense subset of f−1[{0}]. Additionally, if Intf−1[{0}]=0̸, each x∈{0,1}∩E(f) is an accumulation point of E(f). Our main result states that, for a closed nowhere dense set F⊂[0,1] with each x∈{0,1}∩F being an accumulation point of F, there exists f∈C∞[0,1] such that F=E(f)=f−1[{0}].
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GF20-22230L" target="_blank" >GF20-22230L: Banach spaces of continuous and Lipschitz functions</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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