Darboux integral

In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral. Moreover, the definition is readily extended to defining Riemann–Stieltjes integration. Darboux integrals are named after their inventor, Gaston Darboux (1842–1917). The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function f {\displaystyle f} on the interval [ a , b ] . {\displaystyle [a,b].} The Darboux integral exists if and only if the upper and lower integrals are equal. The upper and lower integrals are in turn the infimum and supremum, respectively, of upper and lower (Darboux) sums which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of f in each subinterval of the partition. These ideas are made precise below: A partition of an interval [ a , b ] {\displaystyle [a,b]} is a finite sequence of values x i {\displaystyle x_{i}} such that Each interval [ x i − 1 , x i ] {\displaystyle [x_{i-1},x_{i}]} is called a subinterval of the partition.