Darboux Sums of Sums, Scalar Multiples and Dominated Functions, and the Oscillation Bound

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$. A map $h:[a,b]\to\mathbb{R}$ is bounded if there is $B\in\mathbb{R}$ with $|h(t)|\le B$ for every $t\in[a,b]$. Let $f,g:[a,b]\to\mathbb{R}$ be bounded maps, and let the sum $f+g$ and the scalar multiple $cf$, for $c\in\mathbb{R}$, be as in Pointwise Operations on Real-Valued Maps (i), with $I=[a,b]$. Let $P$ be a partition of $[a,b]$, let $n\in\mathbb{N}$ and the increasing listing $a=x_{1}<x_{2}<\dots<x_{n+1}=b$ of $P$ be as in The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (i), with $x_{i}$ and $x_{i+1}$ defined for $i\in[n]$ by The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (ii), and, for a bounded map $h:[a,b]\to\mathbb{R}$, let $U(h,P)$ and $L(h,P)$ be the upper sum and the lower sum of $h$ with respect to $P$.

(i) $f+g$ is bounded, $U(f+g,P)\le U(f,P)+U(g,P)$ and $L(f,P)+L(g,P)\le L(f+g,P)$.

(ii) Let $c\in\mathbb{R}$. Then $cf$ is bounded. If $c\ge0$, then $U(cf,P)=c\,U(f,P)$ and $L(cf,P)=c\,L(f,P)$. If $c<0$, then $U(cf,P)=c\,L(f,P)$ and $L(cf,P)=c\,U(f,P)$.

(iii) Let $c\in\mathbb{R}$. If $f(t)\le g(t)+c$ for every $t\in[a,b]$, then $U(f,P)\le U(g,P)+c\,(b-a)$ and $L(f,P)\le L(g,P)+c\,(b-a)$.

(iv) Let $\eta\in\mathbb{R}$ with $\eta\ge0$. If $|f(s)-f(t)|\le\eta$ for every $i\in[n]$ and all $s,t\in[x_{i},x_{i+1}]$, then $U(f,P)-L(f,P)\le\eta\,(b-a)$.