Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$, and let $f:[a,b]\to\mathbb{R}$ be bounded: there is $B\in\mathbb{R}$ with $|f(t)|\le B$ for every $t\in[a,b]$. Let partitions of $[a,b]$ and their refinements be as in Partitions of a Closed Interval, the Set of Partitions, and Refinements, let $U(f,P)$ and $L(f,P)$ be the upper sum and the lower sum of $f$ with respect to a partition $P$; they lie in $\mathbb{R}$ by Upper and Lower Darboux Sums of a Bounded Function on a Partition (ii) and Upper and Lower Darboux Sums of a Bounded Function on a Partition (iii). Let $P$ and $Q$ be partitions of $[a,b]$.

(i) If $\ell,C\in\mathbb{R}$ satisfy $\ell\le f(t)\le C$ for every $t\in[a,b]$, then

$$\ell\,(b-a)\le L(f,P)\le U(f,P)\le C\,(b-a).$$

(ii) If $Q$ is a refinement of $P$, then $L(f,P)\le L(f,Q)\le U(f,Q)\le U(f,P)$.

(iii) $L(f,P)\le U(f,Q)$.

(iv) There is exactly one $\lambda_{*}\in\mathbb{R}$ such that $L(f,R)\le\lambda_{*}$ for every partition $R$ of $[a,b]$, and $\lambda_{*}\le\mu$ for every $\mu\in\mathbb{R}$ with $L(f,R)\le\mu$ for every partition $R$ of $[a,b]$. There is exactly one $\lambda^{*}\in\mathbb{R}$ such that $\lambda^{*}\le U(f,R)$ for every partition $R$ of $[a,b]$, and $\mu\le\lambda^{*}$ for every $\mu\in\mathbb{R}$ with $\mu\le U(f,R)$ for every partition $R$ of $[a,b]$. Moreover, $\lambda_{*}\le\lambda^{*}$.

(v) Let $c\in\mathbb{R}$ with $a<c<b$, let $P_{1}$ be a partition of $[a,c]$ and let $P_{2}$ be a partition of $[c,b]$. Then $P_{1}\cup P_{2}$ is a partition of $[a,b]$, the restrictions $f|_{[a,c]}:[a,c]\to\mathbb{R}$ and $f|_{[c,b]}:[c,b]\to\mathbb{R}$ are bounded, and

$$U(f,P_{1}\cup P_{2})=U(f|_{[a,c]},P_{1})+U(f|_{[c,b]},P_{2}),\qquad L(f,P_{1}\cup P_{2})=L(f|_{[a,c]},P_{1})+L(f|_{[c,b]},P_{2}),$$

where $U(f|_{[a,c]},P_{1})$ and $L(f|_{[a,c]},P_{1})$ are the sums of Upper and Lower Darboux Sums of a Bounded Function on a Partition (ii) and Upper and Lower Darboux Sums of a Bounded Function on a Partition (iii) with $[a,c]$ in place of $[a,b]$, and $U(f|_{[c,b]},P_{2})$ and $L(f|_{[c,b]},P_{2})$ those with $[c,b]$ in place of $[a,b]$.