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A century of Sierpiński–Zygmund functions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Sierpiński–Zygmund (\({{\,\mathrm{SZ}\,}}\)) functions are the maps from \({\mathbb {R}}\) to \({\mathbb {R}}\) that have “as little continuity” as possible. In this work we discuss the history behind their discovery, their constructions through the years, and their generalizations. The presentation emphasizes the algebraic properties of \({{\,\mathrm{SZ}\,}}\) maps and their relation to different classes of generalized continuous-like functions. From the seminal work of Blumberg and the appearance of Sierpiński–Zygmund’s result, we describe the current state of the art of this century-old class of functions and discuss the impact that it has had on several different directions of research. Many typical proofs used in the theory, often in a simplified format never published before, are included in the presented material. Moreover, open problems and new directions of research are indicated.

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Notes

  1. All birth and death dates we include in this work are publicly available.

  2. Krzysztof Chris Ciesielski (1957–), the first author, is a Polish American mathematician. He received his Ph.D. in 1985 from Warsaw University and the same year moved to the USA. Since 1989 he works at West Virginia University (USA) where he directed, so far, five Ph.D. students, two of which, F. Jordan and K. Płotka, contributed to this story. His research is in foundations of mathematics and, since 2004, in image processing. Around 2006 he began adding his middle name, Chris, in his publications.

  3. Current Ph.D. student of J. B. Seoane-Sepúlveda.

  4. Juan Benigno Seoane-Sepúlveda (1978–), the second author, is a Spanish mathematician. He received his first Ph.D. at the Universidad de Cádiz (Spain) jointly with Universität Karlsruhe (Germany) in 2005. His second Ph.D. was earned at Kent State University (Kent, Ohio, USA) in 2006 under the supervision of Profs. Richard M. Aron and Vladimir I. Gurariy (whose work inspired parts of this story). Since 2010 he’s a professor at Universidad Complutense de Madrid (Spain) and has directed five Ph.D. theses.

  5. Indeed, by Lemma 2.5, a function \(f\in {\mathbb {R}}^{\mathbb {R}}\) belongs to \({{\,\mathrm{SZ}\,}}({{\,\mathrm{{\mathscr {C}}}\,}})\) if, and only if, \(|f\cap g|<{\mathfrak {c}}\) for every continuous g from a \(G_\delta \)-set \(G\subset {\mathbb {R}}\) into \({\mathbb {R}}\). Since any such g has an extension \({\hat{g}}\in {{\,\mathrm{{\mathscr {B}}}\,}}\), we have \({{\,\mathrm{SZ}\,}}({{\,\mathrm{{\mathscr {C}}}\,}})\supset {{\,\mathrm{SZ}\,}}({{\,\mathrm{{\mathscr {B}}}\,}})\).

  6. This property is sometimes referred to as being star-like, see e.g. [56].

  7. The largest in a sense of a size of the minimal cardinality of generating set.

  8. The definition of such a family in [57] additionally assumes that each \(S\in {{\mathcal {S}}}\) has cardinality \(\lambda \). We do not impose it here, but apply this definition only to such families. But the distinction is important, since in the model from Theorem 3.3 there are \(2^{\mathfrak {c}}\) many subsets of \(\omega _1\subset \omega _2\) which, according to our definition, are \({\mathfrak {c}}\)-almost disjoint. Nevertheless, (4) from Theorem 3.2 fails in this model.

  9. Darboux made many important contributions to geometry and mathematical analysis. His alma mater was the Ecole Normale Supérieure (in Paris). He was a biographer of Henri Poincaré (1854–1912). In 1908, he was a plenary speaker at the International Congress of Mathematicians in Rome.

  10. Except that we, additionally, get \({{\,\mathrm{{\mathscr {D}}}\,}}\subset {{\,\mathrm{SCIVP}\,}}= {{\,\mathrm{CIVP}\,}}\).

  11. Young was co-founder and a president of the MAA. He was also editor of the Bulletin of the American Mathematical Society.

  12. Nash shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. In 2015, he also shared the Abel Prize with Louis Nirenberg for his work on nonlinear PDEs.

  13. Stallings’ contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six.

  14. By [74, lemma 1], \({{\,\mathrm{dom}\,}}(B)\) has nonempty interior. Thus, by the Baire category theorem, there exists \(n\in {\mathbb {N}}\) for which the same is true for the set \(B_n:=B\cap ({\mathbb {R}}\times [-n,n])\). If J is a nonempty interval contained in \({{\,\mathrm{dom}\,}}(B_n)\) and \(h:J\rightarrow {\mathbb {R}}\) is defined via \(h(x)=\inf \{y:\langle x,y\rangle \in B_n\}\), then h is of Baire class 1. Thus, \(g:=h\upharpoonright C(h)\) is as needed.

  15. The map \(\gamma \) from \(X:= J\cap {{\,\mathrm{dom}\,}}(g)\) into \(g\upharpoonright J\subset {\mathbb {R}}^2\), given as \(\gamma (x):=\langle x, g(x)\rangle \), is continuous and so \(g\upharpoonright J=\gamma [J\cap {{\,\mathrm{dom}\,}}(g)]=\gamma [{\mathrm{cl}}_X(D)]\subset {\mathrm{cl}}_{\mathbb {R}}(\gamma [D])={\mathrm{cl}}_{\mathbb {R}}({\hat{g}}\cap g\upharpoonright J)\).

  16. Current Ph.D. student of Ciesielski.

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Acknowledgements

The authors would like to thank Prof. Dr. Tomasz Natkaniec for his very careful reading of an earlier version of this manuscript, especially for many insightful comments and suggestions. We also like to express our gratitude to Dr. George Blumberg, Dr. Larisa Lev Altshuler, and the Real Analysis Exchange journal for their help in providing the photographs from Figs. 1, 4, and 3 , respectively. J. B. Seoane-Sepúlveda was supported by Grants MTM2015-65825-P and PGC2018-097286-B-I00.

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  1. Department of Mathematics, West Virginia University, Morgantown, WV, 26506-6310, USA

    K. C. Ciesielski

  2. Department of Radiology, MIPG, University of Pennsylvania, Philadelphia, PA, 19104-6021, USA

    K. C. Ciesielski

  3. Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    J. B. Seoane-Sepúlveda

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Ciesielski, K.C., Seoane-Sepúlveda, J.B. A century of Sierpiński–Zygmund functions. RACSAM 113, 3863–3901 (2019). https://doi.org/10.1007/s13398-019-00726-0

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