On the Property N − 1

and Applied Analysis 3 Proof. Let B 1 , B 2 , A, and f be the same as in Theorem 4. It is easily seen that every member of B 2 is of the form 0.i 1 ⋅ ⋅ ⋅ i n 001010 ⋅ ⋅ ⋅ or 0.i 1 ⋅ ⋅ ⋅ i n 1101010 ⋅ ⋅ ⋅ for some n ∈ N, except 1/6, 1/3, 2/3, 5/6. Define φ : {0, 1} ∪ B 1 ∪ B 2 → {0, 1} ∪ B 1 ∪ B 2 by


Introduction
First we will specify some basic notations.By || we denote the Lebesgue measure of  ⊂ R. For any  :  → R, where  is an interval, by  ↾  we denote the restriction of  to  ⊂  and the symbol   ap () stands for approximate derivative of  at . Definition 1 (see [1]).Let  ⊂ R be measurable.We say that  :  → R has Lusin's property (), if the image () of every set  ⊂  of Lebesgue measure 0 has Lebesgue measure 0.
This condition was studied exhaustively; some of results can be found in [1].For the present paper the most important is the following.
In the present paper we will study a similar property.Definition 3 (see [2,3]).We say that  :  → R, defined on a measurable set  ⊂ R, has  −1 -property, if the inverse image  −1 () of every set  ⊂ R of Lebesgue measure 0 has Lebesgue measure 0. Some of results concerning  −1 -property are presented in [2,3].In [2] a systematic study of  −1 -property for smooth and almost everywhere differentiable functions can be found.Some applications of  −1 -property in functional equation and geometric function theory can be found in [4][5][6].

Main Results
Our goal is to construct a continuous function  : [0, 1] → [0, 1] with  −1 -property which is not approximately differentiable on a set of full measure.We start with the basic theorem.
Corollary 5. Any function , defined on an interval, which possesses Lusin's condition () such that the set of discontinuity points of  is finite, is derivable at every point of some set of positive Lebesgue measure.