Proof of Integration by Substitution on a Closed Interval

Proof of (i). Let $u,v\in J$ with $u<v$. Then $[u,v]\subseteq J$ by Basic Facts about Intervals of the Real Line (ii), and $[u,v]$ is an interval containing two distinct points by Basic Facts about Intervals of the Real Line (v). As $f$ is continuous at every point of $J$ relative to $J$, the restriction $f|_{[u,v]}:[u,v]\to\mathbb{R}$ is continuous at every point of $[u,v]$ relative to $[u,v]$ by Restriction Stability of Continuity and of the Derivative (i), applied with $I=J$ and $[u,v]$ in place of $J$, hence continuous on $[u,v]$ by The Real Line: Standing Notation and Background for Calculus on Intervals (vi). By A Continuous Function on a Closed Interval is Riemann Integrable (i), applied to $u<v$ and $f|_{[u,v]}$, this restriction is bounded and integrable on $[u,v]$, which by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi) says that $f$ is integrable on $[u,v]$ and defines $\int_{u}^{v}f$.

Now let $u,v\in J$ be arbitrary. If $u<v$, $\int_{u}^{v}f$ is defined by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi), as just shown. If $u=v$, then $\int_{u}^{u}f=0$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii). If $v<u$, then $f$ is integrable on $[v,u]$ by the first paragraph, and $\int_{u}^{v}f=-\int_{v}^{u}f$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii). In each case $\int_{u}^{v}f$ is defined in the sense of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii).

Proof of (ii). Let $t\in[a,b]$. As $\varphi$ is differentiable at $t$, it is continuous at $t$ by The Increment Form of the Derivative, and Differentiability Implies Continuity (ii); as $t$ was arbitrary, $\varphi$ is continuous on $[a,b]$ by The Real Line: Standing Notation and Background for Calculus on Intervals (vi). By the composition rule of The Real Line: Standing Notation and Background for Calculus on Intervals (vi), applied with $A=[a,b]$, $B=J$ and $f$ in place of $g$, the composition $f\circ\varphi:[a,b]\to\mathbb{R}$ is continuous on $[a,b]$. The map $g$ is the pointwise product $(f\circ\varphi)\,\varphi'$, as both have the value $f(\varphi(t))\,\varphi'(t)$ at every $t\in[a,b]$, so $g$ is continuous on $[a,b]$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v), applied with $(X,d)=(\mathbb{R},d_{\mathbb{R}})$ and $A=[a,b]$. Hence $g$ is bounded by Extreme Value Theorem on a Closed Real Interval (i) and integrable on $[a,b]$ by A Continuous Function on a Closed Interval is Riemann Integrable (i).

The range of $\varphi$. By Extreme Value Theorem on a Closed Real Interval (ii) there are $p,q\in[a,b]$ with $\varphi(p)\le\varphi(t)\le\varphi(q)$ for every $t\in[a,b]$. Let $m=\varphi(p)$ and $M=\varphi(q)$. Then $m,M\in J$ and $m\le M$, so $K=[m,M]\subseteq J$ by Basic Facts about Intervals of the Real Line (ii), and $\varphi(t)\in K$ for every $t\in[a,b]$; in particular $\varphi(a),\varphi(b)\in K$.

Case $m=M$. Then $\varphi(t)=m$ for every $t\in[a,b]$, so $\varphi$ is the constant map with value $m$. By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i), $0$ is the derivative of this map at every $t\in[a,b]$, so $\varphi'(t)=0$, the derivative being unique by Derivative of a Real Function on an Interval (ii). Hence $g(t)=f(m)\cdot0=0$ for every $t\in[a,b]$, so $g$ is the constant map with value $0$, and $\int_{a}^{b}g=0\cdot(b-a)=0$ by Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (vi). On the other side, $\varphi(a)=\varphi(b)=m$, and $\int_{m}^{m}f=0$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii). As $\int_{a}^{b}g$ is $\int_{a}^{b}f(\varphi(t))\,\varphi'(t)\,dt$ in the notation of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v), this is (iii).

Case $m<M$. Then $K$ is an interval containing two distinct points by Basic Facts about Intervals of the Real Line (v). Let $f_{K}=f|_{K}:K\to\mathbb{R}$. By Restriction Stability of Continuity and of the Derivative (i), applied with $I=J$ and $K$ in place of $J$, $f_{K}$ is continuous at every point of $K$ relative to $K$, hence continuous on $K$ by The Real Line: Standing Notation and Background for Calculus on Intervals (vi); so it is bounded by Extreme Value Theorem on a Closed Real Interval (i) and integrable on $K$ by A Continuous Function on a Closed Interval is Riemann Integrable (i). For all $u,v\in K$, Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (iv), applied with $E=J$, $g=f$ and the subset $K$ of $J$, gives (1) below. Indeed, if $u=v$, both sides of (1) are $0$ by its last sentence. If $u<v$, then $[u,v]\subseteq K$ by Basic Facts about Intervals of the Real Line (ii); $f$ is integrable on $[u,v]$ by (i), as $u,v\in J$; so $f_{K}$ is integrable on $[u,v]$, Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (iv) being an equivalence, and its first equation gives $\int_{u}^{v}f_{K}=\int_{u}^{v}f$. If $v<u$, then likewise $[v,u]\subseteq K$ and $f$ and $f_{K}$ are integrable on $[v,u]$, and the second equation of Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (iv), with $v$ and $u$ in place of $u$ and $v$, gives $\int_{u}^{v}f_{K}=\int_{u}^{v}f$. Thus

$$\int_{u}^{v}f_{K}=\int_{u}^{v}f\qquad\text{for all }u,v\in K.\qquad\text{(1)}$$

Let $H:K\to\mathbb{R}$ be the map $H(u)=\int_{m}^{u}f_{K}$, with derivatives taken relative to $K$; it exists by Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (i), applied with $m$, $M$ and $f_{K}$ in place of $a$, $b$ and $f$, and it is the map $F$ there. As $f_{K}$ is continuous at every $u\in K$, Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (iv) shows that $H$ is differentiable at every $u\in K$ with $H'(u)=f_{K}(u)=f(u)$.

Let $t\in[a,b]$. By Chain Rule for Real Functions on Intervals (i), applied with the intervals $[a,b]$ and $K$, with $\varphi$ in place of $f$ (its values lie in $K$) and $H$ in place of $g$, the composition $H\circ\varphi:[a,b]\to\mathbb{R}$ is differentiable at $t$ and

$$(H\circ\varphi)'(t)=H'(\varphi(t))\,\varphi'(t)=f(\varphi(t))\,\varphi'(t)=g(t).$$

By The Increment Form of the Derivative, and Differentiability Implies Continuity (ii), $H\circ\varphi$ is continuous at $t$; as $t$ was arbitrary, $H\circ\varphi$ is continuous on $[a,b]$ by The Real Line: Standing Notation and Background for Calculus on Intervals (vi), and it is differentiable at every point of $(a,b)$ with derivative $g$ there. Since $g$ is bounded and integrable on $[a,b]$ by (ii), Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative (i), applied to $g$ in place of $f$ and $H\circ\varphi$ in place of $F$, gives

$$\int_{a}^{b}g=H(\varphi(b))-H(\varphi(a)).$$

By Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (iii), applied to $f_{K}$ on $[m,M]$ with $u=m$, $v=\varphi(a)$ and $w=\varphi(b)$, we have $\int_{m}^{\varphi(b)}f_{K}=\int_{m}^{\varphi(a)}f_{K}+\int_{\varphi(a)}^{\varphi(b)}f_{K}$, that is, $H(\varphi(b))-H(\varphi(a))=\int_{\varphi(a)}^{\varphi(b)}f_{K}$. By (1) this equals $\int_{\varphi(a)}^{\varphi(b)}f$. As $\int_{a}^{b}g$ is $\int_{a}^{b}f(\varphi(t))\,\varphi'(t)\,dt$ in the notation of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v), this is (iii).