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 man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)