## On the Denjoy-Young-Saks theorem. (Sur le théorème de Denjoy-Young-Saks.)(French)Zbl 0835.26006

Let $$U, F: [a, b]\to \mathbb{R}$$ be functions, where $$U$$ is strictly increasing. Let $$A\subset \mathbb{R}$$ be a nonempty set and let $$r: A\to \mathbb{R}_+$$ be a positive function (a gauge). Denote by $$F(A, r)$$ the set of all classes $$(I_i)_{i\leq k}$$ $$(k\geq 1)$$ of closed intervals $$I_i= [a_i, b_i]$$ such that there are points $$x_i\in I_i\cap A$$ with $$I_i\subset (x_i- r(x_i), x_i+ r(x_i))$$ for $$i\leq k$$ and $$a\leq a_1<b_1\leq\cdots\leq a_k< b_k\leq b$$. For $$P= (I_i)_{i\leq k}\in F(A, r)$$ let $$W_F(P)= \sum^k_{i= 1} |F(b_i)- F(a_i)|$$ and let $$m_F(A)= \inf_r \sup\{W_F(P); P\in F(A, r)\}$$, where for $$r: A\to \mathbb{R}_+$$ we can take an arbitrary gauge. A set $$A\subset [a, b]$$ is said to be an $$F$$-null set if $$A= B\cup C$$, where $$m_F(B)= 0$$ and $$C$$ is countable. It is proved that $$[a, b]= E_1\cup E_2\cup E_3\cup E_4$$, where $$F$$ is $$U$$-differentiable on $$E_1$$ (i.e., for each $$x\in E_1$$ there is $$\lim_{y\to x} ((F(y)- F(x))/ (U(y)- U(x)))\in \mathbb{R})$$, $$\underline D_U F(x)= - \infty$$ and $$\overline D_U F(x)= \infty$$ on $$E_2$$, $$\underline D_U F(x)= \overline D_U F(x)= \pm \infty$$ on $$E_3$$ and $$E_3$$ is a $$U$$-null set and $$E_4$$ is an $$F$$-null set and a $$U$$-null set.

### MSC:

 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 28A12 Contents, measures, outer measures, capacities 28A15 Abstract differentiation theory, differentiation of set functions