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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:
  J. Haussermann, Generalized porosity characteristics of a residual set of continuous functions, Ph. D. dissertation, University of California, Santa Barbara, 1984.  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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