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Concept
Darboux integral
Summary
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).
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:
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.
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.
Official source
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