It is shown to be consistent that every function of first Baire class can be decomposed into continuous functions yet the least cardinal of a dominating family in is . The model used in the one obtained by adding Miller reals to a model of the Continuum Hypothesis.
@article{bwmeta1.element.bwnjournal-article-fmv145i2p171bwm,
author = {Saharon Shelah and Juris Steprans},
title = {Decomposing Baire class 1 functions into continuous functions},
journal = {Fundamenta Mathematicae},
volume = {144},
year = {1994},
pages = {171-180},
zbl = {0821.03022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv145i2p171bwm}
}
Shelah, Saharon; Steprans, Juris. Decomposing Baire class 1 functions into continuous functions. Fundamenta Mathematicae, Tome 144 (1994) pp. 171-180. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv145i2p171bwm/
[00000] [1] J. Cichoń, M. Morayne, J. Pawlikowski, and S. Solecki, Decomposing Baire functions, J. Symbolic Logic 56 (1991), 1273-1283. | Zbl 0742.04003
[00001] [2] M. Groszek, Combinatorics on ideals and forcing with trees, ibid. 52 (1987), 582-593. | Zbl 0646.03048
[00002] [3] A. Miller, Rational perfect set forcing, in: Axiomatic Set Theory, D. A. Martin, J. Baumgartner and S. Shelah (eds.), Contemp. Math. 31, Amer. Math. Soc., Providence, R.I., 1984, 143-159.
[00003] [4] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, Berlin, 1982.
[00004] [5] J. Steprāns, A very discontinuous Borel function, J. Symbolic Logic 58 (1993), 1268-1283. | Zbl 0805.03036