Proof of Term-by-Term Differentiation of a Series of Continuously Differentiable Functions with Summable Uniform Bounds

Each result cited is universally quantified over the data in its own statement. Throughout, $x\in\mathbb{R}^{q}$ and $i\in[q]$ are fixed, $e_{i}$ is the $i$th standard basis vector of $\mathbb{R}^{q}$, and $x+te_{i}$ is the point whose $i$th coordinate is $x_{i}+t$ and whose other coordinates are those of $x$ (Euclidean Points as Tuples of Real Numbers); for $\phi:\mathbb{R}^{q}\to\mathbb{R}$ we write $\phi^{x,i}:[-1,1]\to\mathbb{R}$, $\phi^{x,i}(t)=\phi(x+te_{i})$, for the section of $\phi$ along the $i$th coordinate line through $x$, where $[-1,1]$ is the closed interval and every $t$ with $-1<t<1$ is an interior point of it by Basic Facts about Intervals of the Real Line (v). The map $t\mapsto x+te_{i}$ from $[-1,1]$ to $\mathbb{R}^{q}$ satisfies $d_{E}(x+te_{i},x+t'e_{i})=|t-t'|$ (claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, the difference of the two points having the single nonzero coordinate $t-t'$), hence is continuous; so the section of a continuous $\phi$ is continuous on $[-1,1]$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map. Moreover, if the partial derivative $\partial_{i}\phi(x+t_{0}e_{i})$ exists for some $-1<t_{0}<1$, then $\phi^{x,i}$ is differentiable at $t_{0}$ with derivative $\partial_{i}\phi(x+t_{0}e_{i})$: the defining condition of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ at the point $x+t_{0}e_{i}$, in which the point $(a_{1},\dots,a_{i}+h,\dots,a_{q})$ is $x+(t_{0}+h)e_{i}$, is the defining condition of Derivative of a Real Function on an Interval for $\phi^{x,i}$ at $t_{0}$ once $\delta$ is decreased so that $|h|<\delta$ forces $t_{0}+h\in[-1,1]$, which is possible since $t_{0}$ is interior.

Claim 1. For every $k$ one has $|f_{k}(x)|\le a_{k}$ and $|\partial_{i}f_{k}(x)|\le b_{k}$, and the series $\sum a_{k}$ and $\sum b_{k}$ converge; so $\sum_{k}f_{k}(x)$ and $\sum_{k}\partial_{i}f_{k}(x)$ converge absolutely by An Absolutely Convergent Series of Real Numbers Converges (iii), which also gives $|F(x)|\le\sum_{k}a_{k}$ and $|G_{i}(x)|\le\sum_{k}b_{k}$, the bounds asserted in claim 2.

Claim 2, continuity of the partial sums and of the limits. For $m\in\mathbb{N}$ let $S_{m}=\sum_{k=1}^{m}f_{k}$ and $S^{i}_{m}=\sum_{k=1}^{m}\partial_{i}f_{k}$ be the pointwise finite sums, so that $S_{m}(y)$ and $S_{m}^{i}(y)$ are the partial sums of the two series at $y$. Let $T$ be the set of $m\in\mathbb{N}$ such that $S_{m}$ is of class $C^{1}$ on $\mathbb{R}^{q}$ with $\partial_{i}S_{m}=S^{i}_{m}$ for every $i\in[q]$. Then $1\in T$ since $S_{1}=f_{1}$ and $S^{i}_{1}=\partial_{i}f_{1}$ (claim 1 of Properties of Finite Sums); and if $m\in T$ then $S_{m+1}=S_{m}+f_{m+1}$ (claim 1 of Properties of Finite Sums) is of class $C^{1}$ by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, with $\partial_{i}S_{m+1}=\partial_{i}S_{m}+\partial_{i}f_{m+1}=S^{i}_{m+1}$ by claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions and again claim 1 of Properties of Finite Sums, so $m+1\in T$. By Principle of Induction for the Natural Numbers, $T=\mathbb{N}$. In particular every $S_{m}$ and every $S^{i}_{m}$ is continuous on $\mathbb{R}^{q}$ (a function of class $C^{1}$ is continuous, and $\partial_{i}S_{m}$ is continuous, by clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ read through its clause 3). By The Weierstrass M-Test (iii), applied with $S=\mathbb{R}^{q}$ and the bounds $a_{k}$, respectively $b_{k}$, the partial sums $S_{m}$ converge to $F$ uniformly on $\mathbb{R}^{q}$ and the partial sums $S^{i}_{m}$ converge to $G_{i}$ uniformly on $\mathbb{R}^{q}$; hence $F$ and every $G_{i}$ are continuous on $\mathbb{R}^{q}$ by Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (iii).

Claim 2, the partial derivative of $F$. Put $\phi_{k}=f_{k}^{x,i}$ and $h_{k}=(\partial_{i}f_{k})^{x,i}$, continuous on $[-1,1]$ by the preamble, and $G=G_{i}^{x,i}$, continuous on $[-1,1]$ by the previous paragraph. By the preamble, $\phi_{k}$ is differentiable at every $t$ with $-1<t<1$ with derivative $h_{k}(t)$. Let $\Gamma_{k}:[-1,1]\to\mathbb{R}$, $\Gamma_{k}(t)=\int_{-1}^{t}h_{k}(\tau)\,d\tau$, and $\Gamma:[-1,1]\to\mathbb{R}$, $\Gamma(t)=\int_{-1}^{t}G(\tau)\,d\tau$, be the indefinite Riemann integrals of claim 1 of Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity; by claims 2 and 3 of that theorem $\Gamma_{k}$ and $\Gamma$ are continuous on $[-1,1]$ and differentiable at every $t$ with $-1<t<1$, with $\Gamma_{k}'(t)=h_{k}(t)$ and $\Gamma'(t)=G(t)$.

Step (a): $\phi_{k}(t)-\phi_{k}(0)=\Gamma_{k}(t)-\Gamma_{k}(0)$ for every $t\in[-1,1]$. The function $\Gamma_{k}-\phi_{k}$ is continuous on $[-1,1]$ (claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied to the sum of $\Gamma_{k}$ and the multiple $(-1)\phi_{k}$) and differentiable at every interior point with derivative $h_{k}(t)-h_{k}(t)=0$ by claims 1 and 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers. Let $u<v$ be points of $[-1,1]$. The restriction of $\Gamma_{k}-\phi_{k}$ to $[u,v]$ is continuous on $[u,v]$ (claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map) and differentiable at every $t$ with $u<t<v$ with derivative $0$, the defining condition of Derivative of a Real Function on an Interval for the restriction being implied by that for the function; so by Mean Value Theorem on a Closed Real Interval and on Any Interval there is $c$ with $u<c<v$ and $(\Gamma_{k}-\phi_{k})(v)-(\Gamma_{k}-\phi_{k})(u)=0\cdot(v-u)=0$ (claim 1 of Zero Products and Elementary Identities in a Field). Applying this with $\{u,v\}=\{0,t\}$ for $t\ne0$ gives step (a); for $t=0$ it is trivial.

Step (b): $\Gamma(t)=\sum_{k=1}^{\infty}\Gamma_{k}(t)$ for every $t\in[-1,1]$. For $t=-1$ both sides are $0$ by the degenerate-interval convention in claim 1 of Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity. Let $-1<t\le1$. By claim 3 of Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval, applied on $[-1,t]$ to the restrictions of $h_{k}$ and of $G$, which are continuous on $[-1,t]$ (claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map), one has $\Gamma_{k}(t)=\int_{\mathbb{R}}\tilde{h}_{k}\,d\lambda$ and $\Gamma(t)=\int_{\mathbb{R}}\tilde{G}\,d\lambda$, where $\tilde{h}_{k}$ and $\tilde{G}$ are the zero extensions off $[-1,t]$ and $\lambda$ is Lebesgue measure, these zero extensions being measurable and integrable by claim 2 there. For $m\in\mathbb{N}$ the finite sum $\sum_{k=1}^{m}\tilde{h}_{k}$ is the zero extension of the restriction of $S^{i}_{m}$'s section $(S^{i}_{m})^{x,i}=\sum_{k=1}^{m}h_{k}$ to $[-1,t]$; it converges pointwise on $\mathbb{R}$ to $\tilde{G}$, since $S^{i}_{m}\to G_{i}$ pointwise (claim 1 of Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps with $T=S=\mathbb{R}^{q}$) and both sides vanish off $[-1,t]$; and it is dominated by the function $b\,\mathbf{1}_{[-1,t]}$, $b=\sum_{k}b_{k}$, since $|\sum_{k=1}^{m}h_{k}(\tau)|\le\sum_{k=1}^{m}|h_{k}(\tau)|\le\sum_{k=1}^{m}b_{k}\le b$ by claims 2 and 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (i); the finite sums $\sum_{k=1}^{m}\tilde{h}_{k}$ are measurable by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps. The dominating function is measurable (claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps) and integrable: it is a nonnegative multiple of the indicator of the Borel set $[-1,t]$, whose integral is $b\,\lambda([-1,t])=b(t+1)<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, the integral of an indicator being the measure of the set (Simple Functions, Nonnegative Simple Functions, and the Simple Integral with Nonnegative Measurable Maps and Their Lebesgue Integral) and $\lambda([-1,t])=t+1$ by claim 4 of Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness. By Dominated Convergence Theorem, $\int_{\mathbb{R}}\sum_{k=1}^{m}\tilde{h}_{k}\,d\lambda\to\int_{\mathbb{R}}\tilde{G}\,d\lambda=\Gamma(t)$; and by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (induction on $m$ as in the previous paragraph) the left side equals $\sum_{k=1}^{m}\Gamma_{k}(t)$, the $m$th partial sum of $\sum_{k}\Gamma_{k}(t)$. Hence that series converges with sum $\Gamma(t)$, by Series of Real Numbers (ii).

Step (c): conclusion. For $t\in[-1,1]$, by Series of Real Numbers (ii) and Elementary Properties of Series of Real Numbers (i), step (a) and step (b),

$$F(x+te_{i})-F(x)=\sum_{k=1}^{\infty}\bigl(\phi_{k}(t)-\phi_{k}(0)\bigr)=\sum_{k=1}^{\infty}\bigl(\Gamma_{k}(t)-\Gamma_{k}(0)\bigr)=\Gamma(t)-\Gamma(0).$$

Thus $F^{x,i}(t)=\Gamma(t)+\bigl(F(x)-\Gamma(0)\bigr)$ on $[-1,1]$, and by claim 1 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (sum with a constant function, whose derivative is $0$, at the interior point $0$) $F^{x,i}$ is differentiable at $0$ with derivative $\Gamma'(0)=G(0)=G_{i}(x)$. By the identification in the preamble, read in the converse direction (the defining condition of Derivative of a Real Function on an Interval for $F^{x,i}$ at $0$ with $\delta\le1$ is the defining condition of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ at $x$, the point $x+he_{i}$ lying in the open set $\mathbb{R}^{q}$), the partial derivative $\partial_{i}F(x)$ exists and equals $G_{i}(x)$. As $x$ and $i$ were arbitrary, $\partial_{i}F=G_{i}$ on $\mathbb{R}^{q}$, and $F$ is continuous with continuous partial derivatives, hence of class $C^{1}$ on $\mathbb{R}^{q}$ by clauses 1 and 3 of Functions of Class Cᵏ on an Open Subset of ℝⁿ. The bounds were obtained in claim 1.