Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.For all ε > 0, there exists δ > 0 such that for any tagged partition x0, ..., xn and t0, ..., tn − 1 whose mesh is less than δ, we haveFor all ε > 0, there exists a tagged partition y0, ..., ym and r0, ..., rm − 1 such that for any tagged partition x0, ..., xn and t0, ..., tn − 1 which is a refinement of y0, ..., ym and r0, ..., rm − 1, we haveOne direction can be proven using the oscillation definition of continuity: For every positive ε, Let Xε be the set of points in with oscillation of at least ε. Since every point where f is discontinuous has a positive oscillation and vice versa, the set of points in , where f is discontinuous is equal to the union over {X1/n} for all natural numbers n. In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral. Let f be a non-negative real-valued function on the interval , and let be the region of the plane under the graph of the function f and above the interval (see the figure on the top right). We are interested in measuring the area of S. Once we have measured it, we will denote the area by: The basic idea of the Riemann integral is to use very simple approximations for the area of S. By taking better and better approximations, we can say that 'in the limit' we get exactly the area of S under the curve. Note that where f can be both positive and negative, the definition of S is modified so that the integral corresponds to the signed area under the graph of f: that is, the area above the x-axis minus the area below the x-axis. A partition of an interval is a finite sequence of numbers of the form Each is called a sub-interval of the partition. The mesh or norm of a partition is defined to be the length of the longest sub-interval, that is, A tagged partition P(x, t) of an interval is a partition together with a finite sequence of numbers t0, ..., tn − 1 subject to the conditions that for each i, ti ∈ . In other words, it is a partition together with a distinguished point of every sub-interval. The mesh of a tagged partition is the same as that of an ordinary partition.