# Sierpiński's hierarchy and locally Lipschitz functions

Fundamenta Mathematicae (1995)

• Volume: 147, Issue: 1, page 73-82
• ISSN: 0016-2736

top

## Abstract

top
Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and $\alpha <{\omega }_{1}$ then f ○ g ∈ ${B}_{\alpha }\left(Z\right)$ for every $g\in {B}_{\alpha }\left(Z\right){\cap }^{Z}I$ if and only if f is continuous on I, where ${B}_{\alpha }\left(Z\right)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes ${S}_{\alpha }\left(Z\right)\left(\alpha >0\right)$ in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz on I (thus it holds for the class of differences of semicontinuous functions, which is the class ${S}_{1}\left(Z\right)$). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside ${S}_{\alpha }\left(Z\right)$ by outer superpositions.

## How to cite

top

Morayne, Michał. "Sierpiński's hierarchy and locally Lipschitz functions." Fundamenta Mathematicae 147.1 (1995): 73-82. <http://eudml.org/doc/212075>.

@article{Morayne1995,
abstract = {Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and $α < ω_1$ then f ○ g ∈ $B_α(Z)$ for every $g ∈ B_α(Z) ∩^ZI$ if and only if f is continuous on I, where $B_α(Z)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes $S_α(Z) (α > 0)$ in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz on I (thus it holds for the class of differences of semicontinuous functions, which is the class $S_1(Z)$). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside $S_α(Z)$ by outer superpositions.},
author = {Morayne, Michał},
journal = {Fundamenta Mathematicae},
keywords = {locally Lipschitz functions; Sierpiński classification of Borel measurable functions},
language = {eng},
number = {1},
pages = {73-82},
title = {Sierpiński's hierarchy and locally Lipschitz functions},
url = {http://eudml.org/doc/212075},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Morayne, Michał
TI - Sierpiński's hierarchy and locally Lipschitz functions
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 1
SP - 73
EP - 82
AB - Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and $α < ω_1$ then f ○ g ∈ $B_α(Z)$ for every $g ∈ B_α(Z) ∩^ZI$ if and only if f is continuous on I, where $B_α(Z)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes $S_α(Z) (α > 0)$ in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz on I (thus it holds for the class of differences of semicontinuous functions, which is the class $S_1(Z)$). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside $S_α(Z)$ by outer superpositions.
LA - eng
KW - locally Lipschitz functions; Sierpiński classification of Borel measurable functions
UR - http://eudml.org/doc/212075
ER -

## References

top
1. [CM1] J. Cichoń and M. Morayne, Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), 83-89. Zbl0646.26009
2. [CM2] J. Cichoń and M. Morayne, An abstract version of Sierpiński's theorem and the algebra generated by A and CA functions, Fund. Math. 142 (1993), 263-268. Zbl0824.54009
3. [H] F. Hausdorff, Set Theory, Chelsea, New York, 1962.
4. [Ke] S. Kempisty, Sur les séries itérées des fonctions continues, Fund. Math. 2 (1921), 64-73. Zbl48.0276.04
5. [Ku] K. Kuratowski, Topology I, Academic Press, New York and London, 1966.
6. [L] A. Lindenbaum, Sur les superpositions de fonctions représentables analytiquement, Fund. Math. 23 (1934), 15-37; Corrections, ibid., 304. Zbl60.0195.02
7. [Mau] R. D. Mauldin, Baire functions, Borel sets, and ordinary function systems, Adv. in Math. 12 (1974), 418-450. Zbl0278.26005
8. [Maz] S. Mazurkiewicz, Sur les fonctions de classe 1, Fund. Math. 2 (1921), 28-36.
9. [Mor] M. Morayne, Algebras of Borel measurable functions, ibid. 141 (1992), 229-242. Zbl0812.26004
10. [S1] W. Sierpiński, Sur les fonctions développables en séries absolument convergentes de fonctions continues, ibid. 2 (1921), 15-27. Zbl48.0276.01
11. [S2] W. Sierpiński, Démonstration d'un théorème sur les fonctions de première classe, ibid. 2 (1921), 37-40. Zbl48.0276.03

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.