Finite representation of continuous functions, Nina Bary's wrinkled functions and Foran's condition (M)
Real Analysis Exchange 15 (1990), 445-469.
Mathematical reviews subject classification: 26A30.
Abstract
Nina Bary shows (see Mémoire sur la representation finie des fonctions continues, Math. Ann., 103 (1930), 185-248 and 598-653) the following chain of inclusions:
quasi-derivable ⊂ S+S ⊂ C = S+S+S
for continuous functions on [0,1].
It can be shown that above, Banach's condition S can be replaced by
GE(1) ∩ T1 ⊂ S,
where GE(1) is defined using condition E(1) of Article 4.
In the present paper we define the conditions
GAC*D*1 ⊂ GAC*D1 ⊆ GE(1) ∩ T1
for continuous functions on [0,1], with which we prove the above results.
To prove that S+S ≠ C, Nina Bary introduced the wrinkled functions W (she called them fonctions ridées) and showed that W is not empty.
In this paper we give characterizations of the wrinkled functions, which show that between Foran's condition (M) (introduced in 1979 in the article A generalization of absolute continuity, Real Analysis Exchange 5 (1979/80), 82-91) and these functions there is a very close relationship.
So we improve Nina Bary's results on wrinkled functions.
Finally, we construct a wrinkled function which is approximately derivable at no point of [0,1], and for which each level set is perfect.
