The ℑ-density topology on ℝ is a refinement of the natural topology. It is a category analogue of the density topology [9, 10]. This paper is concerned with ℑ-density continuous functions, i.e., the real functions that are continuous when the ℑ-densitytopology is used on the domain and the range. It is shown that the family of ordinary continuous functions f: [0,1]→ℝ which have at least one point of ℑ-density continuity is a first category subset of C([0,1])= f: [0,1]→ℝ: f is continuous equipped with the uniform norm. It is also proved that the class of ℑ-density continuous functions, equipped with the topology of uniform convergence, is of first category in itself. These results remain true when the ℑ-density topology is replaced by the deep ℑ-density topology.
@article{bwmeta1.element.bwnjournal-article-fmv140i1p79bwm,
author = {K. Ciesielski and L. Larson},
title = {Category theorems concerning Z-density continuous functions},
journal = {Fundamenta Mathematicae},
volume = {138},
year = {1991},
pages = {79-85},
zbl = {0807.54031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv140i1p79bwm}
}
Ciesielski, K.; Larson, L. Category theorems concerning Z-density continuous functions. Fundamenta Mathematicae, Tome 138 (1991) pp. 79-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv140i1p79bwm/
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