The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $p,q\in\mathbb{N}$. Differentiability, derivatives and derivative maps of maps from an interval containing two distinct points into $\mathbb{R}^{q}$ or into $\mathcal{M}_{p\times q}(\mathbb{R})$ are as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i), Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii), Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iv), derivatives being taken relative to that interval; integrability and integrals of such maps are as in Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv). Continuity refers to $d_{\mathbb{R}}$ on the domain and to $d_{E}$ on $\mathbb{R}^{q}$ and $d_{p\times q}$ on $\mathcal{M}_{p\times q}(\mathbb{R})$, as in Continuous Map Between Metric Spaces (ii); $\lVert\cdot\rVert$ is the Euclidean norm on vectors and the operator norm on matrices. Let $I$ be an interval containing two distinct points, as in Single-Variable Calculus on an Interval (ii); a point $t\in I$ is an inner point of $I$ if $u<t<v$ for some $u,v\in I$.

(i) Let $a,b\in\mathbb{R}$ with $a<b$, let $x:[a,b]\to\mathbb{R}^{q}$ be continuous on $[a,b]$, and let $c\in[a,b]$. Then $\int_{c}^{t}x$ is defined for every $t\in[a,b]$, and the map $X:[a,b]\to\mathbb{R}^{q}$, $t\mapsto\int_{c}^{t}x$, is differentiable, with $X'(t)=x(t)$ for every $t\in[a,b]$. The same holds for a continuous $A:[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$ in place of $x$.

(ii) Let $x:I\to\mathbb{R}^{q}$ be differentiable, and let $u,v\in I$ be such that $\int_{u}^{v}x'$ is defined. Then

$$\int_{u}^{v}x'=x(v)-x(u).$$

The same holds for a differentiable $A:I\to\mathcal{M}_{p\times q}(\mathbb{R})$ in place of $x$.

(iii) Let $x:I\to\mathbb{R}^{q}$ be differentiable, with derivative $x'$ continuous on $I$. Then, for all $u,v\in I$, $\int_{u}^{v}x'$ is defined and

$$\int_{u}^{v}x'=x(v)-x(u).$$

The same holds for a differentiable $A:I\to\mathcal{M}_{p\times q}(\mathbb{R})$ with $A'$ continuous on $I$, in place of $x$.

(iv) Let $x:I\to\mathbb{R}^{q}$ be differentiable with $x'(t)=0$ for every $t\in I$. Then $x(u)=x(v)$ for all $u,v\in I$. The same holds for a differentiable $A:I\to\mathcal{M}_{p\times q}(\mathbb{R})$ with $A'(t)=0_{p\times q}$ for every $t\in I$.

(v) Let $x:I\to\mathbb{R}^{q}$ be continuous on $I$ and differentiable at every inner point of $I$, and let $M\in\mathbb{R}$ satisfy $\lVert x'(t)\rVert\le M$ for every inner point $t$ of $I$. Then

$$\lVert x(v)-x(u)\rVert\le M\,|v-u|\qquad(u,v\in I).$$

Likewise, if $A:I\to\mathcal{M}_{p\times q}(\mathbb{R})$ is continuous on $I$ and differentiable at every inner point of $I$, and $\lVert A'(t)\rVert\le M$ for every inner point $t$ of $I$, then $\lVert A(v)-A(u)\rVert\le M\,|v-u|$ for all $u,v\in I$.