# 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) ∩ T_{1} ⊂ 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*D_{1} ⊆ GE(1) ∩ T_{1}

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.