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.**