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Section 24.4 Integrability conditions and properties
Objectives
Use upper and lower sums to test whether a bounded function is integrable.
Apply linearity and basic inequalities to analyze integrals.
Explain why continuous functions on compact intervals are always integrable.
Let \(f:[a,b]\to\bb R\) be bounded. Then \(f\) is integrable if and only if for all \(\eps>0\) there exists a partition \(P_{\eps}\) of \([a,b]\) such that
\begin{equation*}
U(f,P_{\eps})-L(f,P_{\eps})<\eps\text{.}
\end{equation*}
If \(f:[a,b]\to\bb R\) is integrable, then there exists a sequence of partitions of \([a,b]\) for which
\begin{equation*}
\limit{n\to\infty}[U(f,P_{n})-L(f,P_{n})]=0\text{.}
\end{equation*}
Continuous functions are integrable. The proof relies on uniform continuity, which is why compactness and uniform continuity were covered in this chapter.
\(\displaystyle \abs{\int f}\le\int |f|\)
