Darboux Integral - Definition

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

\begin{align} M_i = \sup_{x\in} f(x), \\ m_i = \inf_{x\in} f(x) . \end{align}

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.

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