Darboux-like properties of generalized derivatives (Q1068223)

The author gives the following three assertions about path derivative without any intersection assumption for the systems E of paths: 1. If the paths of E are bilateral, if f is continuous or a VBG-function and E- differentiable, then \(f'_ E\) has the Darboux property. 2.If the paths of E are nonporous, if f is continuous or a VBG-function, E- differentiable and if \(f'_ E\) is in the Baire class one, then holds the implication: \[ \{x:f'_ E(x)=\lambda \}\neq \emptyset \Rightarrow \{x:f'_ E(x)=\lambda \}\cap \{x:f'(x)\quad exists\}\neq \emptyset. \] 3. If the paths of E are nonporous, if f is continuous and E-differentiable and if \(f'_ E\) is in the Baire class one, then for each interval Y on which \(f'_ E\) attains both values M and -M there exists such an interval I that \(I\subset Y,\) \(f'_ E=f'\) on I and f' attains both values M and -M on I. There are also some applications of these results.