Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative

This lemma adopts the setting of Single-Variable Calculus on an Interval. For a subset $E\subseteq\mathbb{R}$, a map $g:E\to\mathbb{R}$ and $u,v\in E$, integrability of $g$ on a closed interval, the oriented integral $\int_{u}^{v}g$, and what it means that $\int_{u}^{v}g$ is defined, are as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii). For maps $f,g:E\to\mathbb{R}$ and $c\in\mathbb{R}$, $f+g$ and $cf$ are the sum and scalar multiple of Pointwise Operations on Real-Valued Maps (i), with $E$ for its index set, and in an integral over points of $E$, $c$ also denotes the constant map $E\to\mathbb{R}$ with value $c$. Continuity on a subset is as in Single-Variable Calculus on an Interval (iii); differentiability on an interval and the derivative $f'$ of a differentiable map $f$ are as in Derivative of a Real Function on an Interval (iii) and Derivative of a Real Function on an Interval (iv).

(i) Let $E\subseteq\mathbb{R}$, let $f,g:E\to\mathbb{R}$ and let $u,v\in E$. If $\int_{u}^{v}f$ and $\int_{u}^{v}g$ are defined, then $\int_{u}^{v}(f+g)$ is defined and $\int_{u}^{v}(f+g)=\int_{u}^{v}f+\int_{u}^{v}g$.

(ii) Let $E\subseteq\mathbb{R}$, let $f:E\to\mathbb{R}$, let $u,v\in E$ and let $c\in\mathbb{R}$. If $\int_{u}^{v}f$ is defined, then $\int_{u}^{v}cf$ is defined and $\int_{u}^{v}cf=c\int_{u}^{v}f$.

(iii) Let $E\subseteq\mathbb{R}$, let $c\in\mathbb{R}$, and let $u,v\in E$ with $[u,v]\subseteq E$ if $u<v$ and $[v,u]\subseteq E$ if $v<u$. Then $\int_{u}^{v}c$ is defined and $\int_{u}^{v}c=(v-u)\,c$.

(iv) Let $I$ be an interval containing two distinct points and $f:I\to\mathbb{R}$ a map, as in Single-Variable Calculus on an Interval (ii), and let $f$ be continuous on $I$. Then $\int_{u}^{v}f$ is defined for all $u,v\in I$.

(v) Let $I$ be an interval containing two distinct points and $f:I\to\mathbb{R}$ a map continuous on $I$. Then, for all $u,v,w\in I$, the integrals below are defined and

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

(vi) Let $I$ be an interval containing two distinct points, let $f:I\to\mathbb{R}$ be differentiable on $I$, with derivative $f':I\to\mathbb{R}$, and let $u,v\in I$ be such that $\int_{u}^{v}f'$ is defined. Then $\int_{u}^{v}f'=f(v)-f(u)$.