Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions

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 $t\in[a,b]$. Integrability on $[a,b]$ and the integral $\int_{a}^{b}f$ are as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v), integrability on a closed subinterval $[u,v]$ of $[a,b]$ and the integral $\int_{u}^{v}f$ as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi), and the oriented integrals $\int_{u}^{v}f$ for $u,v\in[a,b]$ with $v\le u$ as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii).

(i) Let $c\in\mathbb{R}$ with $a<c<b$. Then $f$ is integrable on $[a,b]$ if and only if $f$ is integrable on $[a,c]$ and on $[c,b]$, and in that case

$$\int_{a}^{b}f=\int_{a}^{c}f+\int_{c}^{b}f .$$

(ii) If $f$ is integrable on $[a,b]$, then $f$ is integrable on $[u,v]$ for all $u,v\in[a,b]$ with $u<v$.

(iii) If $f$ is integrable on $[a,b]$, then for all $u,v,w\in[a,b]$,

$$\int_{u}^{w}f=\int_{u}^{v}f+\int_{v}^{w}f .$$

(iv) Let $K\subseteq E\subseteq\mathbb{R}$, let $g:E\to\mathbb{R}$, and let $g|_{K}:K\to\mathbb{R}$ be its restriction, with integrability on closed intervals and the integrals of $g$ and of $g|_{K}$ as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii). For $u,v\in K$ with $u<v$ and $[u,v]\subseteq K$, $g|_{K}$ is integrable on $[u,v]$ if and only if $g$ is integrable on $[u,v]$, and in that case $\int_{u}^{v}g|_{K}=\int_{u}^{v}g$ and $\int_{v}^{u}g|_{K}=\int_{v}^{u}g$. For every $u\in K$, $\int_{u}^{u}g|_{K}=\int_{u}^{u}g=0$.