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
and in Article 4 we defined the condition E(N) and the class
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:
and ![\small G_N \notin {\mathcal B}(N^2+2N) \small G_N \notin {\mathcal B}(N^2+2N)](http://latex.codecogs.com/gif.latex?%5Csmall%20G_N%20%5Cnotin%20%7B%5Cmathcal%20B%7D%28N%5E2%2B2N%29)
We also construct a continuous function which satisfies Foran's condition N and which is not in
.