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

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals.

(i) Derivative. For every interval $I$ containing two distinct points, every function $f:I\to\mathbb{R}$, every $x\in I$ and every $L\in\mathbb{R}$, $L$ is a derivative of $f$ at $x$ in the earlier version if and only if it is a derivative of $f$ at $x$ in the new version.

(ii) For $I$, $f$ and $x$ as in (i), $f$ is differentiable at $x$ in the earlier version if and only if it is differentiable at $x$ in the new version, and then $f'(x)$ is the same real number in both.

(iii) For $I$ and $f$ as in (i) and every $S\subseteq I$, $f$ is differentiable on $S$ in the earlier version if and only if it is differentiable on $S$ in the new version.

(iv) For $I$ and $f$ as in (i), if $f$ is differentiable, its derivative $f'$ in the earlier version is its derivative in the new version.

(v) Riemann integral. For all $a,b\in\mathbb{R}$ with $a<b$ and every bounded function $f:[a,b]\to\mathbb{R}$, the lower and upper integrals of $f$ over $[a,b]$ in the earlier version are its lower and upper integrals in the new version; $f$ is integrable on $[a,b]$ in the earlier version if and only if it is integrable on $[a,b]$ in the new version.

(vi) For $a$, $b$ and $f$ as in (v), if $f$ is integrable on $[a,b]$, its Riemann integral in the earlier version is its Riemann integral in the new version.

(vii) For every $E\subseteq\mathbb{R}$, every function $g:E\to\mathbb{R}$ and all $u,v\in\mathbb{R}$ with $u<v$ and $[u,v]\subseteq E$, $g$ is integrable on $[u,v]$ in the sense of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi) if and only if it is in the sense of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi), and then $\int_{u}^{v}g$ is the same real number in both.

(viii) For $E$ and $g$ as in (vii) and all $u,v\in E$, the integral $\int_{u}^{v}g$ is defined in the sense of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii) if and only if it is defined in the sense of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii); and then the two versions, through 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), and through 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), give the same real number $\int_{u}^{v}g$.