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.


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