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Article 13

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.