Abstract
Suppose {Mn} is a sequence of pairwise disjoint, nowhere dense closed subsets of [0, 1] and {Fn} is a sequence of continuous functions. We show that there exists a continuous function F which has the same derivate structure as Fn at each point of Mn. In addition, F can be made BV if ∑n=1∞ V(Fn, Mn), the sum of the variation of Fn|Mn, is finite. A well-known and very useful theorem of Laczkovich and Petruska as well as many classical examples follow readily from our results.
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Darji, U.B. Two General Extension Theorems. Acta Mathematica Hungarica 83, 97–106 (1999). https://doi.org/10.1023/A:1006667620395
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DOI: https://doi.org/10.1023/A:1006667620395