I've been working through a book on Introductory Real Analysis and I've been stumped by part of this problem.
Suppose we consider a partition that splits $[a,b]$ into $n$ partitions each with length $\frac{b-a}{n}$. Show that a function $f:[a,b]\rightarrow\mathbb R$ is Darboux integrable on just these partitions iff it is Darboux integrable over all partitions.
The if direction is very trivial, that one isn't too tricky.
However, the only if direction has me stumped. How can I show that integrability over this type of partition implies integrability over any partitions. I think I can show that it gives integrability over rational partitions (not sure if that's a formal term, basically a partition where all interval lengths are rational). However, I'm having trouble extending it to all partitions in general.
Best Answer
Let $\mathcal{P}$ ($\mathcal{P}_U$) be the collection of all partitions (uniform partitions) of $[a,b]$.
For a bounded function $f$, we have $L(P,f) \leqslant U(Q,f)$ for lower and upper Darboux sums corresponding to arbitrary partitions $P$ and $Q$. It follows that
$$\sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}}U(P,f), \quad \sup_{P \in \mathcal{P}_U}L(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)$$
Since $\mathcal{P}_U \subset \mathcal{P}$ we have
$$\{U(P,f) \,|\, P \in \mathcal{P}_U\} \subset \{U(P,f) \,|\, P \in \mathcal{P}\}, \quad \{L(P,f) \,|\, P \in \mathcal{P}_U\} \subset \{L(P,f) \,|\, P \in \mathcal{P}\}$$
Hence,
$$\tag{*}\sup_{P \in \mathcal{P}_U}L(P,f) \leqslant \sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}}U(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)$$
Darboux integrability with respect to uniform partitions means that
$$\sup_{P \in \mathcal{P}_U}L(P,f) =\inf_{P \in \mathcal{P}_U}U(P,f),$$
which, in view of (*), implies that $f$ is Darboux integrable with respect to all partitions since
$$0 \leqslant \inf_{P \in \mathcal{P}}U(P,f)- \sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)- \sup_{P \in \mathcal{P}_U}L(P,f) = 0,$$ and $$ \inf_{P \in \mathcal{P}}U(P,f)= \sup_{P \in \mathcal{P}}L(P,f) $$