Fractional Order Continuity and Some Properties about Integrability and Differentiability of Real Functions*

In this paper a certain function space C , 0 1, larger than the space of continuous functions, is introduced in order to study new properties and the extension of some already known results about the Riemann–Liouville fractional integral and derivative operators. Sufficient conditions for the continuity of I1− a f are given. Furthermore, necessary conditions are given for the pointwise existence of fractional derivatives. The existence of a derivative of order , from the existence of a certain derivative of order , < , is also analyzed.