In the branch of mathematics known as 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). Definition The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function on the interval 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: Darboux sums A partition of an interval is a finite sequence of values xi such that Each interval is called a subinterval of the partition. Let be a bounded function, and let be a partition of . Let The upper Darboux sum of with respect to is The lower Darboux sum of with respect to is The lower and upper Darboux sums are often called the lower and upper sums. Darboux integrals The upper Darboux integral of f is The lower Darboux integral of f is In some literature an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively. and like Darboux sums they are sometimes simply called the lower and upper integrals. If Uf = Lf, then we call the common value the Darboux integral. We also say that f is Darboux-integrable or simply integrable and set An equivalent and sometimes useful criterion for the integrability of f is to show that for every ε