On the Cesaro and Borel symmetric derivatives

The paper discusses some properties of Cesaro and Borel derivatives. It was established that, according to Lebesgue, the concepts of Cesaro and Borel first-order symmetric (ordinary) derivatives are equivalent for a summable function. The dependence has been studied of the ordinary and symmetric Borel and Cesaro derivatives on the set of positive. In particular, if is a set of any positive measure, and at every point of this set there exists a symmetric Cesaro (respectively Borel) derivative, then almost everywhere on this set, there exists a Cesaro (respectively Borel) derivative equal to the symmetric derivative. It is also shown that a summable function can have an infinitely large Cesaro (respectively Borel) symmetric derivative on a set of zero measure.