## 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

### MSC:

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