# On Foran's conditions A(N), B(N) and (M)

### Real Analysis Exchange 9 (1984), 495-502

#### Mathematical reviews subject classification: 26A45, 26A46, 26A39

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## Abstract

Two continuous functions F_{1} and F_{2} 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 F_{1} and F_{2} do not differ by a constant.

It is also shown that G(x) = F_{2}(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.