Generalizations of Cesàro continuous functions and integrals of Perron type

Abstract

The linear space of all the Cesàro continuous functions of any order is extended by introducing pointwisely Cesàro continuous functions and exact generalized Peano derivatives. Then six generalized integrals of Perron type are defined and studied. They are based on three recent monotonicity theorems and each depends on an abstract upper semilinear space of certain functions. Some of the integrals are more general than all the integrals in the Cesàro-Perron scale provided that the abstract semilinear space is taken to be the linear space of all the pointwisely Cesàro continuous functions or all the exact generalized Peano derivatives. That such a concrete general integral is possible follows from the fact proved here that each exact generalized Peano derivative is in Baire class one and has the Darboux property. Relations between the pointwisely Cesàro continuous functions or the exact generalized Peano derivatives and functions defined by means of the values of certain Schwartz’s distributions at "points" are also established.