# \(\mathcal I\)-Density Continuous Functions

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*Krzysztof Ciesielski; Lee Larson; Krzysztof M Ostaszewski*

The classical approach to showing the parallel between theorems concerning Lebesgue measure and theorems concerning Baire category on the real line is restricted to sets of measure zero and sets of first category. This is because classical Baire category theory does not have an analogue for the Lebesgue density theorem. By using \({\mathcal I}\)-density, this deficiency is removed, and much of the structure of measurable sets and functions can be shown to exist in the sense of category as well. This monograph explores category analogues to such things as the density topology, approximate continuity, and density continuity. In addition, some questions about topological semigroups of real functions are answered.

#### Table of Contents

# Table of Contents

## $\mathcal I$-Density Continuous Functions

- Contents vii8 free
- Introduction xi12 free
- Chapter 1. The Ordinary Density Topology 116 free
- Chapter 2. Category Analogues of the Density Topology 1732
- 2.1. J-density and J-dispersion Points 1732
- 2.2. I-density and I-dispersion Points 2237
- 2.3. The I-density Topology 2843
- 2.4. The P*-topology 2944
- 2.5. I-approximate Continuity 3348
- 2.6. Topological Properties of the I-density Topology 3752
- 2.7. The Deep-I-density Topology 3954
- 2.8. I-density Topologies Versus the Density Topology 4257
- 2.9. Historical and Bibliographic Notes 4459

- Chapter 3. I-density Continuous Functions 4762
- 3.1. I-density and Deep-I-density Continuous Functions 4762
- 3.2. Homeomorphisms and I-density 4964
- 3.3. Addition within H ∩ C[sub(II)] 5671
- 3.4. More I-density Continuous Functions 6883
- 3.5. I-density Continuous Functions are Baire*1 7388
- 3.6. Inclusions and Examples 7893
- 3.7. I-density Versus Density Continuous Functions 89104
- 3.8. Other Continuities 93108
- 3.9. Historical and Bibliographic Notes 99114

- Chapter 4. Semigroups 101116
- Appendix A. Notation 123138
- References 127142
- Index 131146