Editing Darboux integral (section)

==Definition==
The definition of the Darboux integral considers '''upper and lower (Darboux) integrals''', which exist for any [[bounded function|bounded]] [[real number|real]]-valued function <math>f</math> on the [[interval (mathematics)|interval]] <math>[a,b].</math> 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===
[[Image:Darboux.svg|thumb|right|Lower (green) and upper (green plus lavender) Darboux sums for four subintervals|350x350px]]
A [[partition of an interval]] <math>[a,b]</math> is a finite sequence of values <math>x_{i}</math> such that

:<math>a = x_0 < x_1 < \cdots < x_n = b.</math>

Each interval <math>[x_{i-1},x_i]</math> is called a ''subinterval'' of the partition. Let <math>f:[a,b]\to\R</math> be a bounded function, and let

:<math>P = (x_0, \ldots, x_n)</math>

be a partition of <math>[a,b]</math>. Let

:<math>\begin{align}
M_i = \sup_{x\in[x_{i-1},x_{i}]} f(x), \\
m_i = \inf_{x\in[x_{i-1},x_{i}]} f(x).
\end{align}</math>


The '''upper Darboux sum''' of <math>f</math> with respect to <math>P</math> is

:<math>U_{f, P} = \sum_{i=1}^n (x_{i}-x_{i-1}) M_i. \,\!</math>

The '''lower Darboux sum''' of <math>f</math> with respect to <math>P</math> is

:<math>L_{f, P} = \sum_{i=1}^n (x_{i}-x_{i-1}) m_i. \,\!</math>

The lower and upper Darboux sums are often called the lower and upper sums.

===Darboux integrals===
The '''upper Darboux integral''' of ''f'' is

:<math>U_f = \inf\{U_{f,P} \colon P \text{ is a partition of } [a,b]\}.</math>

The '''lower Darboux integral''' of ''f'' is

:<math>L_f = \sup\{L_{f,P} \colon P \text{ is a partition of } [a,b]\}.</math>

In some literature, an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively:

:<math>\begin{align}
&{} L_f \equiv \underline{\int_{a}^{b}} f(x) \, \mathrm{d}x, \\
&{} U_f \equiv \overline{\int_{a}^{b}} f(x) \, \mathrm{d}x,
\end{align}</math>

and like Darboux sums they are sometimes simply called the ''lower and upper integrals''.

If ''U''<sub>''f''</sub>&nbsp;=&nbsp;''L''<sub>''f''</sub>, then we call the common value the ''Darboux integral''.<ref>Wolfram MathWorld</ref> We also say that ''f'' is ''Darboux-integrable'' or simply ''integrable'' and set

:<math>\int_a^b {f(t)\,dt} = U_f = L_f.</math>

An equivalent and sometimes useful criterion for the integrability of ''f'' is to show that for every ε > 0 there exists a partition ''P''<sub>ε</sub> of [''a'',&thinsp;''b''] such that<ref>Spivak 2008, chapter 13.</ref>

:<math>U_{f,P_\epsilon} - L_{f,P_\epsilon} < \varepsilon.</math>