Let HN and HI stand for the increasing homeomorphisms that are density and I-density continuous and let HN-1 and HI-1 denote the classes of inverses of functions from HN and HI, respectively; i.e., classes of increasing homeomorphisms that preserve density and I-density points. In the paper we prove that classes HN, HI, and HI-1 are closed under the addition operation. A similar result for the class HN-1 has been proved by Niewiarowski. The theorem that the class HI-1 is closed under the addition operation is also contained in the paper of Aversa and Wilczynski [Homeomorphisms preserving I-density points, Boll. Un. Mat. Ital. B(7)1:275--285, 1987]. However, their proof contains an essential gap. (The gap is discussed in the last paragraph of the paper.)
This paper contains also the examples showing that none of the above theorems is correct if we admit the possibility that one of the homeomorphism is increasing, and the second one is decreasing, even in the case when their sum is still a homeomorphism.
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Last modified April 29, 1999.