An extension of the ordinary variation
\textit{J. Foran} [Proc. Am. Math. Soc. 49, 359-365 (1975; Zbl 0305.26004)] has introduced condition B(N) which for \(N=1\) is identical to the condition of bounded variation. Using condition B(N) we introduce the variation \(V_ N(F;E)\) of a function F on a set E which for \(N=1\) is identical to the ordinary variation of F on E. Then we show that there exist functions F on \([0,1]\) for which \(V_ 2(F;[0,x]\cap C)=\phi (x)\) \((C=Cantor's\) ternary set, \(\phi =Cantor's\) ternary function, \(x\in C).\) Using this new variation we show that there exist continuous functions, satisfying Lusin's condition (N) on \([0,1],\) which are B(N) on C for no natural number N.
scientific article; zbMATH DE number 4029863