Real Analysis Exchange

More about Sierpiński-Zygmund uniform limits of extendable functions

Harvey Rosen

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Let $SZ$, $D$, $Ext$, and $\overline{Ext}$ denote respectively the spaces of Sierpiński-Zygmund functions, Darboux functions, extendable connectivity functions, and uniform limits of sequences of extendable connectivity functions, with the metric of uniform convergence on them. We show that the subspaces $SZ \cap D$ and $SZ \cap \overline{Ext}$ are each porous in the space $SZ$, but $SZ \cap \overline{Ext}$ is not porous in the space $\overline{Ext}$. We also show that every real function can be expressed as a sum of two Sierpiński-Zygmund functions one of which belongs to $\overline{Ext}$. Ciesielski and Natkaniec showed in 1997 that if $\R$ is not the union of less than $\mathfrak c$-many nowhere dense subsets, then there exist Sierpiński-Zygmund bijections $f,g:\mathbb{R}\to\mathbb{R}$ such that $f^{-1} \notin SZ$ and $g^{-1}\in SZ$, but here we can additionally have $f$ and $g$ belonging to $\overline{Ext}$

Article information

Real Anal. Exchange, Volume 30, Number 1 (2004), 129-136.

First available in Project Euclid: 27 July 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 54C35: Function spaces [See also 46Exx, 58D15]

Sierpiński-Zygmund function Darboux function uniform limit of extendable connectivity functions porosity inverse function


Rosen, Harvey. More about Sierpiński-Zygmund uniform limits of extendable functions. Real Anal. Exchange 30 (2004), no. 1, 129--136.

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