Extreme path derivatives of a function with respect to a nonporous system of paths (Q1119758)

The paper deals with the classification of the extreme path-derivatives of a real function of a real variable if the system of paths is nonporous. A system of paths \(E=\{E_ x:\quad x\in R\}\) is said to be nonporous iff the right and left porosities of \(E_ x\) at x are zero. If \(f:\quad R\to R\) is a function and \(E=\{E_ x:\quad x\in R\}\) is a system of paths, the extreme E-derivatives of f at x are the limits \[ \limsup_{y\to x,\quad y\in E_ x}\frac{f(y)-f(x)}{y-x}\quad and\quad \liminf_{y\to x,\quad y\in E_ x}\frac{f(y)-f(x)}{y-x}. \] If these two limits are equal, we say that f has at x an E-derivative \(f_ E'(x).\) The author begins with examples of two functions and two nonporous systems of paths. The first function is Baire 1, E-differentiable and its E-derivative is not Lebesgue measurable, the second function is continuous, E-differentiable and its E-derivative is not Borel measurable. If f: \(<0,1>\to R\) is a function, then an increasing function \(\omega:\quad (0,\infty)\to (0,\infty)\) with the condition \(\lim_{x\to 0+}\omega (x)=0\) will be called a modulus of continuity of f and f belongs to the class C(\(\omega)\) iff every \(x,y\in <0,1>\) implies \(| f(x)-f(y)| <\omega (| x-y|).\) The modulus of continuity \(\omega\) will be regular iff \(\omega (\lambda x)\leq (\lambda +1)\omega (x)\) whenever \(0<\lambda\) and \(x(0,\infty)\). Let be \[ \omega_ f'(\delta)=\sup \{| f(x)-f(y)|:\quad x,y\in <0,1>\quad and\quad | x-y| \leq \delta \} \] for any \(\delta >0\). Then \(\omega_ f'\) is a regular modulus of continuity of f. A function f is of the class \(Lip_ M^{\alpha}\) iff \(\omega_ f'(\delta)\leq M\delta^{\alpha}\) for any \(\delta >0\). The main result of the paper is the Theorem 3.4.: Let \(E=\{E_ x:\) \(x\in <0,1>\}\) be a bilateral system of paths and \(\{a_ n\}^{\infty}_{n=1}\) be a decreasing sequence of positive numbers tending to zero so that \(E_ x\cap <x-a_ n,\quad x-a_{n+1}>\) and \(E_ x\cap <x+a_{n+1},x+a_ n>\) are nonempty for each \(x\in <0,1>.\) Let f be of the class C(\(\omega)\). If the following holds: \(\lim_{n\to \infty}\omega (a_ n)(1/a_{n+1}-1/a_ n)=0\) and \(\lim_{n\to \infty}\omega (a_ n-a_{n+1})/a_{n+1}=0\), then it holds: a) If f has the E-derivative \(f_ E'\), the \(f_ E'\in B_ 1\). b) The extreme E- derivatives \(\bar f_ E'\) and \b{f}\({}_ E'\) are in \(B_ 2.\) Corollary. Let f be a Lipschitz function of the class \(\alpha\), \(0<\alpha <1\), and E be a system of paths such that \(E_ x\cap (x- 1/n^{\alpha},x-1/(n+1)^{\alpha})\) and \(E_ x\cap (x+1/(n+1)^{\alpha},\quad x+1/n^{\alpha})\) are nonempty, then it holds: \(1.\quad f_ E'\in B_ 1\) whenever f is E-differentiable. 2. \(\bar f_ E'\) and \b{f}\({}_ E'\) belong to \(B_ 2\).