Proof of The Earlier and the New Versions of the Definitions of the Derivative on an Interval and of the Riemann Integral Agree

The two versions. Each new version treated here has the text of its earlier version, with the same references, the setting included, except for its opening sentence, where Derivative of a Real Function on an Interval and The Riemann Integral on a Closed Interval via Lower and Upper Integrals add that $\mathbb{R}$ is the set of real numbers. The objects introduced there, $I$, $f$ and $x$ for the derivative, and $a$, $b$ and $f$ for the integral, are therefore the same in both versions, and the real numbers meant are in both the elements of $\mathbb{R}$, with the operations and the order of $\mathbb{R}$ (The Real Numbers as a Type (i), The Real Numbers as a Type (ii)). So each defining condition below, with its $\varepsilon$ and $\delta$, its difference quotient, and its suprema and infima, reads identically in both versions, and it remains to check, clause by clause, that what is built from these conditions agrees.

Derivatives. For (i), Derivative of a Real Function on an Interval (i) and Derivative of a Real Function on an Interval (i) both call $L\in\mathbb{R}$ a derivative of $f$ at $x$ when for every real $\varepsilon>0$ there is a real $\delta>0$ with

$$\Bigl|\frac{f(t)-f(x)}{t-x}-L\Bigr|<\varepsilon \qquad (t\in I,\ 0<|t-x|<\delta).$$

This is one condition on $L$, so $L$ is a derivative of $f$ at $x$ in one version exactly when it is in the other.

Differentiability and the value of the derivative. For (ii), differentiability at $x$ is in both versions the existence of a derivative of $f$ at $x$ (Derivative of a Real Function on an Interval (ii), Derivative of a Real Function on an Interval (ii)); since the derivatives at $x$ are the same numbers in both versions by (i), $f$ is differentiable at $x$ in one version exactly when it is in the other. In that case $f'(x)$ is in each version the derivative of $f$ at $x$, and a number with the displayed property is unique by Uniqueness of the Derivative of a Real Function on an Interval (i); as the two versions have the same such numbers, their $f'(x)$ is the same real number.

Differentiability on a set. For (iii), differentiability on $S\subseteq I$ is in both versions differentiability at every point of $S$ (Derivative of a Real Function on an Interval (iii), Derivative of a Real Function on an Interval (iii)), and this agrees pointwise by (ii).

The derivative map. For (iv), let $f$ be differentiable, which by (iii) with $S=I$ means the same in both versions. The derivatives of Derivative of a Real Function on an Interval (iv) and Derivative of a Real Function on an Interval (iv) are maps $I\to\mathbb{R}$ with value $f'(u)$ at each $u\in I$, and these values agree by (ii); so the two maps are equal by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv).

Lower and upper sums. For (v), let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be bounded, a hypothesis that reads the same in both versions. Both versions form the set of lower sums and the set of upper sums,

$$\mathcal{L}(f)=\{L(f,R):R\in\Pi_{a,b}\},\qquad\mathcal{U}(f)=\{U(f,R):R\in\Pi_{a,b}\},$$

from the same lower and upper Darboux sums over the same set $\Pi_{a,b}$ of partitions of $[a,b]$. By the membership descriptions in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (i) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (i), a set lies in the set of lower sums of one version exactly when it is $L(f,R)$ for some partition $R$ of $[a,b]$, that is, exactly when it lies in that of the other; so the two sets of lower sums are equal, and likewise the two sets of upper sums, by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (ii) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (ii).

Lower and upper integrals, integrability. The lower integral is in both versions the supremum in $\mathbb{R}$ of the set of lower sums, and the upper integral the infimum in $\mathbb{R}$ of the set of upper sums (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iii), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iii)). As these sets are equal, so are their suprema and infima, which gives the first part of (v). Integrability on $[a,b]$ is in both versions the equality of the lower and the upper integral (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv)), and these two numbers are the same in both versions, so integrability agrees.

The integral. For (vi), if $f$ is integrable on $[a,b]$, its integral is in both versions the common value of the lower and the upper integral (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v)), which is the same number in both by the preceding paragraph.

Restrictions. For (vii), let $g:E\to\mathbb{R}$ and $u<v$ with $[u,v]\subseteq E$. Both versions call $g$ integrable on $[u,v]$ when the restriction $g|_{[u,v]}$, the same map in both, is bounded and integrable on $[u,v]$, and both set $\int_{u}^{v}g=\int_{u}^{v}g|_{[u,v]}$ (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 (vi)). Boundedness is the same condition in both, and integrability of $g|_{[u,v]}$ and its integral agree by (v) and (vi) applied to $g|_{[u,v]}$ on $[u,v]$; so integrability of $g$ on $[u,v]$ and the number $\int_{u}^{v}g$ agree.

Defined integrals. For (viii), let $u,v\in E$. In both versions $\int_{u}^{v}g$ is defined when $u=v$, or $u<v$, $[u,v]\subseteq E$ and $g$ is integrable on $[u,v]$, or $v<u$, $[v,u]\subseteq E$ and $g$ is integrable on $[v,u]$ (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii)); each alternative is the same condition in both versions, by (vii) for the integrability. The value is then given in both versions by 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 (vii) together with the restriction clauses: it is $0$ when $u=v$, it is $\int_{u}^{v}g$ when $u<v$, and it is $-\int_{v}^{u}g$ when $v<u$. In the last two cases the integral over the closed interval is the same number in both versions by (vii), so the two versions give the same real number $\int_{u}^{v}g$.