Symmetric selective derivatives. (English) Zbl 0643.26004

Let \(I=(0,1)\) and s be a selection for I, that means, that s is a real function defined on the set of all subintervals of I satisfying \(x<s((x,y))<y\) for any \(0<x<y<1.\) The author defines that a real function f defined on I has a symmetric selective derivative at x iff the following limit \[ \lim_{\delta \to 0+}\frac{f(s((x,x+\delta)))-f(s((x- \delta,x)))}{s((x,x+\delta))-s\quad ((x-\delta,x))} \] exists and then he puts sym sf’(x) equal to this limit.
It is proved: There exists a real function f defined on I such that for any derivative h on I there exists a selection s for I such that \(sym sf'(x)=h(x)\) for each \(x\in I.\)
It is easy to see that there exists a real function f defined on I and two selections \(s_ 1\) and \(s_ 2\) for I such that \(\{x\in I: sym s_ 1f'(x)\neq sym s_ 2f'(x)\}=I.\) The author asks whether sym sf’ has more suitable properties if f is a continuous or Darboux Baire one function on I.
Reviewer: L.Mišík


26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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