Level sets of density continuous functions

Ciesielski, Krzysztof; Larson, Lee

doi:10.1090/S0002-9939-1992-1149970-2

Level sets of density continuous functions
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by Krzysztof Ciesielski and Lee Larson PDF

Proc. Amer. Math. Soc. 116 (1992), 963-969 Request permission

Abstract:

A function $f:{\mathbf {R}} \to {\mathbf {R}}$ is density continuous if it is continuous when both its range and domain are endowed with the density topology. The level sets of density continuous functions are characterised as those sets which are density closed and ambiguous.

References

K. Ciesielski and L. Larson, The space of density continuous functions, Acta Math. Hungar. 58 (1991), no. 3-4, 289–296. MR 1153484, DOI 10.1007/BF01903959
Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski, Density continuity versus continuity, Forum Math. 2 (1990), no. 3, 265–275. MR 1050409, DOI 10.1515/form.1990.2.265
Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski, Differentiability and density continuity, Real Anal. Exchange 15 (1989/90), no. 1, 239–247. MR 1042539, DOI 10.2307/44152001
Casper Goffman, C. J. Neugebauer, and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497–505. MR 137805
Richard J. O’Malley, Baire* $1$, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187–192. MR 417352, DOI 10.1090/S0002-9939-1976-0417352-5
Krzysztof Ostaszewski, Continuity in the density topology, Real Anal. Exchange 7 (1981/82), no. 2, 259–270. MR 657324, DOI 10.2307/44153412
Krzysztof Ostaszewski, Semigroups of density-continuous functions, Real Anal. Exchange 14 (1988/89), no. 1, 104–114. MR 988359, DOI 10.2307/44153632
H. W. Pu and H. H. Pu, Associated sets of Baire$^{\ast }$ $1$ functions, Real Anal. Exchange 8 (1982/83), no. 2, 479–485. MR 700198, DOI 10.2307/44153419