Editing Darboux integral (section)

==Properties==
*For any given partition, the upper Darboux sum is always greater than or equal to the lower Darboux sum. Furthermore, the lower Darboux sum is bounded below by the rectangle of width (''b''−''a'') and height inf(''f'') taken over [''a'',&thinsp;''b'']. Likewise, the upper sum is bounded above by the rectangle of width (''b''−''a'') and height sup(''f'').
*:<math>(b-a)\inf_{x \in [a,b]} f(x) \leq L_{f,P} \leq U_{f,P} \leq (b-a)\sup_{x \in [a,b]} f(x)</math>
*The lower and upper Darboux integrals satisfy
*:<math>\underline{\int_{a}^{b}} f(x) \, dx \leq \overline{\int_{a}^{b}} f(x) \, dx</math>
*Given any ''c'' in (''a'',&thinsp;''b'')
*:<math>\begin{align}
\underline{\int_{a}^{b}} f(x) \, dx &= \underline{\int_{a}^{c}} f(x) \, dx + \underline{\int_{c}^{b}} f(x) \, dx\\[6pt]
\overline{\int_{a}^{b}} f(x) \, dx &= \overline{\int_{a}^{c}} f(x) \, dx + \overline{\int_{c}^{b}} f(x) \, dx
\end{align}</math>
*The lower and upper Darboux integrals are not necessarily linear. Suppose that ''g'':[''a'',&thinsp;''b''] → '''R''' is also a bounded function, then the upper and lower integrals satisfy the following inequalities:
*:<math>\begin{align}
\underline{\int_{a}^{b}} f(x) \, dx + \underline{\int_{a}^{b}} g(x) \, dx &\leq \underline{\int_{a}^{b}} (f(x) + g(x)) \, dx\\[6pt]
\overline{\int_{a}^{b}} f(x) \, dx + \overline{\int_{a}^{b}} g(x) \, dx &\geq \overline{\int_{a}^{b}} (f(x) + g(x)) \, dx
\end{align}</math>
*For a constant ''c'' ≥ 0 we have
*:<math>\begin{align}
\underline{\int_{a}^{b}} cf(x) \, dx &= c\underline{\int_{a}^{b}} f(x)\, dx \\[6pt]
\overline{\int_{a}^{b}} cf(x) \, dx &= c\overline{\int_{a}^{b}} f(x)\, dx
\end{align}</math>
*For a constant ''c'' ≤ 0 we have
*:<math>\begin{align}
\underline{\int_{a}^{b}} cf(x)\, dx &= c\overline{\int_{a}^{b}} f(x)\, dx \\[6pt]
\overline{\int_{a}^{b}} cf(x)\, dx &= c\underline{\int_{a}^{b}} f(x)\, dx
\end{align}</math>
*Consider the function
*:<math>\begin{align}
&{} F : [a, b] \to \R \\
&{} F(x) = \underline{\int_{a}^{x}} f(t) \, dt,
\end{align}</math>
:then ''F'' is [[Lipschitz continuous]]. An identical result holds if ''F'' is defined using an upper Darboux integral.