An Extension of Integrals

Extension of Denjoy–Perron–Henstock–Kurzweil integral has been done by replacing the derivative with the approximate derivative or the distributional derivative. However even in these integrals, for instance, it is impossible to calculate the integral of \(f(x) = \frac{1}{x}\). Therefore we need to extend integrations in another direction. The continuity of integral hinders the extension of integrals. In this paper, we propose an extension of integrals that relaxes the continuity of integral and describe properties of this integral. Lastly, we consider applications.