A theorem on level lines of continuous functions |
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Authors: | Z Waksman J Wasilewsky |
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Institution: | (1) Ben Gurion University of the Negev, Be’er Sheva, Israel;(2) Tel Aviv University, Tel Aviv, Israel |
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Abstract: | A property of a continuous functionf(x), x ∈ E
2, similar to the classical intermediate value property is established. Namely, let a Jordan compactJ ⊂ E
2 be the domain of definition off. Then, for each parametrizationx(t), 0≦t≦T,x(0)=x(T), of the boundary Fr(J) ofJ there exists a unique real λ and a unique connected component Φ of the level set {x ∈ J: f(x)=λ} with the following property: any connected subset Ω ofJ containing “opposite” points of Fr(J) (i.e. pointsx(t′) andx(t″) such thatt″−t′=T/2) has a common element with Φ. |
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