The son of a Lutheran pastor, Riemann forsook an initial interest in theology to study mathematics in Berlin and then in Gottingen. He completed his training for the doctorate in 1851 at the latter university, under the guidance of the legendary Carl Gauss. Riemann returned to Gottingen three years later as a lowly unpaid tutor, working his way up the academic ladder to a full professorship in 1859. Yet his teaching career was tragically brief. He fell ill with tuberculosis and spent his last years in Italy, where he died in 1866, only 39 years of age. Although he published only a few papers, his name is attached to a variety of topics in several branches of mathematics: Riemann surfaces, Cauchy-Riemann equations, the Riemann zeta function, Riemannian (that is, non-Euclidean) geometry, and the still-unproven Riemann Hypothesis.
The view that integration was simply a process reverse to differentiation prevailed until the nineteenth century. The familiar conception of the definite integral as the limit of approximating sums was given by Riemann in a paper he submitted upon joining the faculty at Göttingen in 1854. It was not published until 13 years later, and then only after his untimely death. His formulation of what today is known as the "Riemann integral" runs thus:
If f(x) is a continuous function on the interval [a,b] and
where xk * is an arbitrary point in the subinterval [xk-1,xk] and d is the maximum of the lengths of the subintervals.
(This is a modification of Cauchys definition, in which the xk were taken to be the left-endpoints of the subintervals [xk-1,xk].) Riemann subsequently applied his version of the integral to discontinuous functions, producing a remarkable example of an integrable function having infinitely many discontinuities. With this, the study of discontinuous functions gained mathematical legitimacy.