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Darboux integral - Definition | | Darboux integral - Definition: Encyclopedia II - Darboux integral - Definition | | A partition of an interval [a,b] is a finite sequence
Each [xi − 1,xi] is called a subinterval of the partition. A refinement of the partition
is a partition
such that for every i with
there is an integer r(i) such that
xi = y ...
See also:Darboux integral, Darboux integral - Definition, Darboux integral - Facts about the Darboux integral | | | Darboux integral, Darboux integral - Definition, Darboux integral - Facts about the Darboux integral, Riemann integral, Lebesgue integration | | |
| | | Darboux integral: Encyclopedia II - Darboux integral - Definition
Darboux integral - Definition
A partition of an interval [a,b] is a finite sequence
Each [xi − 1,xi] is called a subinterval of the partition. A refinement of the partition
is a partition
such that for every i with
there is an integer r(i) such that
xi = yr(i)
In other words, to make a refinement, one cuts the subintervals into smaller pieces and does not remove any cuts.
Let be a bounded function, and let
be a partition of [a,b]. Let:
The upper Darboux sum of f with respect to P is
The lower Darboux sum of f with respect to P is
The upper Darboux integral of f is
The lower Darboux integral of f is
If Uf = Lf, then we say that f is Darboux-integrable and set to be the common value of the upper and lower Darboux integrals.
Other related archivesDefinitions of mathematical integration, Gaston Darboux, Lebesgue integration, Riemann integral, Riemann integrals, algebra, functions, integral, mathematical limits, mathematics, partition of an interval, real analysis
 Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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