Editing Darboux integral (section)

===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>