Cauchy's residue theorem for a class of real valued functions

Let [a, b] be an interval in a"e Rand let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] \ E to the derivative f, where E is a subset of [a, b] at whose points F can take values +/-infinity or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt integral(a) (b) f = F(b) - F(a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.