Definition
A partition of an interval is a finite sequence of values xi such that
Each interval is called a subinterval of the partition. Let ƒ:→R be a bounded function, and let
be a partition of . Let
The upper Darboux sum of ƒ with respect to P is
The lower Darboux sum of ƒ with respect to P is
The upper Darboux integral of ƒ is
The lower Darboux integral of ƒ is
If Uƒ = Lƒ, then we say that ƒ is Darboux-integrable and set
the common value of the upper and lower Darboux integrals.
Read more about this topic: Darboux Integral
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)