Two general extension theorems (Q1307415)

Consider a sequence \( \{ M_n \} \) of pairwise disjoint, nowhere dense, closed subsets of \([0,1]\) and a sequence \( \{ F_n \} \) of continuous functions. It is shown that there exists a continuous function \(F\) which has the same derivate structure as \( F_n \) at each point of \( M_n \). In addition, if the sum of the variation of \( F_n | M_n \) is finite, \(F\) can be made a function of bounded variation. A very useful theorem of \textit{G. Petruska} and \textit{M. Laczkovich} [Acta Math. Acad. Sci. Hung. 25, 189-212 (1974; Zbl 0279.26003)] follows readily from these results.