Finite Sums of Real-Valued Maps: Continuity, Integral and Derivative

In the setting of Single-Variable Calculus on an Interval, let $J$ be a finite set; sums over $J$ are the sums over a finite set of real numbers. For a set $D$, a family $(f_{j})_{j\in J}$ of maps $D\to\mathbb{R}$ is a map $j\mapsto f_{j}$ from $J$ to the set $\mathbb{R}^{D}$ of maps from $D$ to $\mathbb{R}$ of Sets and Maps: Ordinary Notation (v), and $\sum_{j\in J}f_{j}$ denotes the map $D\to\mathbb{R}$, $u\mapsto\sum_{j\in J}f_{j}(u)$. For $n\in\mathbb{N}$ and $a:[n]\to\mathbb{R}$, $\sum_{j=1}^{n}a_{j}=\sum_{j\in[n]}a_{j}$, whether the left side is read as in Iterated Operations: Finite Sums and Finite Products (i) or as in Sums and Products over a Finite Set and over an Interval (iv), by Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v); so, with $J=[n]$, the clauses below apply to sums $\sum_{j=1}^{n}$.

(i) Let $(X,d)$ be a metric space, $S\subseteq X$ and $s_{0}\in S$, and let $(f_{j})_{j\in J}$ be a family of maps $S\to\mathbb{R}$ each of which is continuous at $s_{0}$ relative to $S$. Then $\sum_{j\in J}f_{j}$ is continuous at $s_{0}$ relative to $S$.

(ii) Let $J\neq\emptyset$, let $E\subseteq\mathbb{R}$ and $u,v\in E$, let $j\mapsto c_{j}$ be a map from $J$ to $\mathbb{R}$, and let $(f_{j})_{j\in J}$ be a family of maps $E\to\mathbb{R}$ such that $\int_{u}^{v}f_{j}$ is defined for every $j\in J$. Then the integral from $u$ to $v$ of the map $E\to\mathbb{R}$, $t\mapsto\sum_{j\in J}c_{j}f_{j}(t)$, is defined, and

$$\int_{u}^{v}\sum_{j\in J}c_{j}f_{j}(t)\,dt=\sum_{j\in J}c_{j}\int_{u}^{v}f_{j}.$$

(iii) Let $t\in I$, and let $(f_{j})_{j\in J}$ be a family of maps $I\to\mathbb{R}$ each of which is differentiable at $t$, with derivative $f_{j}'(t)$. Then $\sum_{j\in J}f_{j}$ is differentiable at $t$, with derivative $\sum_{j\in J}f_{j}'(t)$.