Functions defined by sequences of integrals and the inversion of approximate derived numbers

have been determined.t It is easy to construct sequences for which the limit on the left side of (1) exists and is different from the right side. In the present paper necessary and sufficient conditions are obtained for the existence of this limit in terms of the sequence s,(x). It turns out that the limit function F(x) is independent of f(x). In fact if F(x) is an arbitrary continuous function, there exists a sequence of summable functions sn(x) tending to zero everywhere, for which