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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 GN which satisfies E(N+1) on a perfect set and which is E(N) on no portion of this set.

Moreover GN 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}.