On Foran's conditions A(N), B(N) and (M)
Real Analysis Exchange 9 (1984), 495-502
Mathematical reviews subject classification: 26A45, 26A46, 26A39
Abstract
Two continuous functions F1 and F2 are constructed, satisfying Lusin's condition (N) and Foran's condition B(2) on C (C = the Cantor ternary set), which satisfy Foran's condition A(N) on no portion of C and for no natural number N.
Moreover F'1(x) = F'2(x) a.e. on [0,1], but F1 and F2 do not differ by a constant.
It is also shown that G(x) = F2(x) -½φ(x) (φ = Cantor's ternary function), fulfils Foran's condition (M), but does not fulfil Lusin's condition (N).
Such a function was already obtained by Foran in An extension of the Denjoy integral (Proc. Amer. Math. Soc., 49 (1975), 359-365), but in a more complicated way.
