Sierpiński's hierarchy and locally Lipschitz functions
Fundamenta Mathematicae, Tome 147 (1995) no. 1, pp. 73-82
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.
@article{10_4064_fm_147_1_73_82,
author = {Micha{\l} Morayne},
title = {Sierpi\'nski's hierarchy and locally {Lipschitz} functions},
journal = {Fundamenta Mathematicae},
pages = {73--82},
year = {1995},
volume = {147},
number = {1},
doi = {10.4064/fm-147-1-73-82},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm-147-1-73-82/}
}
Michał Morayne. Sierpiński's hierarchy and locally Lipschitz functions. Fundamenta Mathematicae, Tome 147 (1995) no. 1, pp. 73-82. doi: 10.4064/fm-147-1-73-82
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