# (83, 64)-Konfigurationen in Laguerre-, Mobius-und weiteren by Benz W.

By Benz W.

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We also introduce and investigate the functions which preserve I-density and deep-Idensity points. Let H be the class of all homeomorphisms from R (or any subinterval) to R (or any subinterval). It will be shown that the inclusion CII ⊂ CDD is proper (even in the class of continuous functions) and that H ⊂ CDD . However, ﬁrst we present the following theorem. 1. Let f : R → R be such that f −1 (E) ∈ I for every E ∈ I. Then f is deep-I-density continuous if, and only if, f is I-density continuous.

A point a ∈ R is said to be a deep-I-density point of a set A ∈ B if, and only if, there exists a closed set F ⊂ A ∪ {a} such that a is an I-density point of F . A point a ∈ R is said to be a deep-I-dispersion point of an A ∈ B if, and only if, it is a deep-I-density point of Ac . Similarly deﬁned are left and right deep-I-density and deep-I-dispersion points. Notice that for a closed set the notions of an I-density point and a deepI-density point coincide. Similarly, the notions of an I-dispersion point and a deep-I-dispersion point coincide for open sets.

Then x is a right I-density point of {w : f (w) > 1/n} and it is apparent that x must be a limit point of A˜n . From this, we see that G is dense in A. 2. Every right I-approximately continuous function is of the ﬁrst Baire class. Proof. Let f be right I-approximately continuous on R. It suﬃces to show that {x : f (x) ≥ 0} is a Gδ set. To do this, for each p ∈ N, let Up = {x : f (x) > −1/p} and, for p, q, r, k ∈ N, deﬁne (17) A(p, q, r, k) = x ∈ R: k−1 k , q q ∩ r (Up − x) = ∅ and q A(p, q, r) = (18) A(p, q, r, k).

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