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Typical continuous functions are virtually nonmonotone. (English) Zbl 0564.26002
If \(\phi\) : (0,1\(>\to (0,1>\) is a continuous function then a subset A of \(<0,1>\) is called bilaterally strongly \(\phi\)-porous iff for each \(x\in A\) there exist two sequences \(\{I_ n\}^{\infty}_{n=1}\) and \(\{J_ n\}^{\infty}_{n=1}\) of intervals such that \(I_ n\subset (x-1/n,x)- A\) and \(J_ n\subset (x,x+1/n)-A\) for every natural n and \[ \lim_{n\to \infty}[\frac{dist(x,I_ n)}{| I_ n|}]=\lim_{n\to \infty}[\frac{dist(x,J_ n)}{| J_ n|}]=0, \] where \(dist(x,I_ n)\) is the distance of x and \(I_ n\) and \(| I_ n|\) is the length of \(I_ n\). Let \({\mathcal K}\) be the class of all constant functions on \(<0,1>\) and \({\mathcal M}\) the class of all monotone functions on \(<0,1>\). In the paper ”On the level set structure of a continuous function” (to appear) B. S. Thomson proved that: For every continuous function \(\phi: (0,1>\to (0,1>\) there exits a set A residual in \(C(<0,1>)\) such that for every \(f\in A\) and every \(a\in {\mathcal K}\) the set \(\{x\in <0,1>: f(x)=a(x)\}\) is bilaterally strongly \(\phi\)-porous. In the paper under review, the authors extend Thomson’s result to the class \({\mathcal M}\). They also prove that such an extension to the class of all absolutely continuous functions on \(<0,1>\) is impossible; especially, they prove the following: For every \(\delta >0\) and every \(f\in C(<0,1>)\) there exists an absolutely continuous function g such that the set \(\{x\in <0,1>: f(x)=g(x)\}\) is not bilaterally strongly \(x^{1+\delta}\)-porous.
Reviewer: L.Mišík

MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A48 Monotonic functions, generalizations
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References:
[1] J. Haussermann, Generalized porosity characteristics of a residual set of continuous functions, Ph. D. dissertation, University of California, Santa Barbara, 1984.
[2] B. S. Thomson, On the level set structure of a continuous function, Classical real analysis (Madison, Wis., 1982) Contemp. Math., vol. 42, Amer. Math. Soc., Providence, RI, 1985, pp. 187 – 190.
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