Article 6

On some classes of continuous functions

Real Analysis Exchange 11 (1986), 452-463.

Mathematical reviews subject classification: 26A30, 26A45, 26A46

Abstract

In An extension of the Denjoy integral Proc. Amer. Math. Soc., 49 (1975), 359-365), Foran introduced the conditions
A(N) , B(N), and the class $\small{\mathcal F}$ and in Article 4 we defined the condition E(N) and the class $\small{\mathcal E}$ for a function on a set for some positive integer N.

In the present paper we construct a continuous function G_N which satisfies E(N+1) on a perfect set and which is E(N) on no portion of this set.

Moreover G_N has also the following properties:

$\small G_N \in {\mathcal F}(N^2+2N+1)$ and $\small G_N \notin {\mathcal B}(N^2+2N)$

We also construct a continuous function which satisfies Foran's condition N and which is not in $\small{\mathcal E}$ .