The fundamental theorem of calculus provides a generic formulation of the inverse relationship between the theories of differentiation and integration; meaning,

which is satisfied in Cauchy integration theory. However, the fundamental theorem of calculus has no meaning if the derivative  of a differentiable function is not Riemann integrable. The purpose of the research is to preset the proprieties which should fulfill the derivative  of a differentiable function in order to satisfy the Fundamental Theorem of Calculus in the Riemann integration theory.