Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, continuity at a point relative to a subset and continuity on a subset are as in Continuous Map Between Metric Spaces (i) and Continuous Map Between Metric Spaces (ii).

(i) Let $(X,d_{X})$, $(Y,d_{Y})$ and $(Z,d_{Z})$ be metric spaces, let $X\times Y$ carry the product metric $d_{X\times Y}$, a metric by The Product Metric is a Metric (i), and let $\pi_{1}:X\times Y\to X$ and $\pi_{2}:X\times Y\to Y$ be the projections of Projection, Evaluation and Currying Maps (i). For every $S\subseteq X\times Y$, the restrictions $\pi_{1}|_{S}:S\to X$ and $\pi_{2}|_{S}:S\to Y$ are continuous on $S$. If moreover $A\subseteq X$ and $g:A\to Z$ is continuous on $A$, then there is exactly one map $k:A\times Y\to Z$ with $k(x,y)=g(x)$ for all $x\in A$ and $y\in Y$, and $k$ is continuous on $A\times Y$.

(ii) Let $(X,d)$ be a metric space, let $S\subseteq X$ and let $p,q\in\mathbb{N}$, with $d_{E}$ on $\mathbb{R}^{q}$ and $d_{p\times q}$ on $\mathcal{M}_{p\times q}(\mathbb{R})$ as in Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (vi). Let $x,y:S\to\mathbb{R}^{q}$ and $A,B:S\to\mathcal{M}_{p\times q}(\mathbb{R})$, and let $-x$, $x-y$, $-A$ and $A-B$ be the maps $u\mapsto-x(u)$, $u\mapsto x(u)-y(u)$, $u\mapsto-A(u)$ and $u\mapsto A(u)-B(u)$ of Maps Formed Pointwise from Vector and Matrix Operations (i). For every point $s_{0}\in S$: if $x$ is continuous at $s_{0}$ relative to $S$, then so is $-x$, and if moreover $y$ is, then so is $x-y$; if $A$ is continuous at $s_{0}$ relative to $S$, then so is $-A$, and if moreover $B$ is, then so is $A-B$. If $x$ is continuous on $S$, then so is $-x$, and if moreover $y$ is, then so is $x-y$; if $A$ is continuous on $S$, then so is $-A$, and if moreover $B$ is, then so is $A-B$.

(iii) Let $a,c,d,b\in\mathbb{R}$ with $a\le c<d\le b$, and let $p,q\in\mathbb{N}$. Let $x:[a,b]\to\mathbb{R}^{q}$ be differentiable, derivatives being taken relative to $[a,b]$, with derivative $x':[a,b]\to\mathbb{R}^{q}$. Then the restriction $x|_{[c,d]}:[c,d]\to\mathbb{R}^{q}$ is differentiable, derivatives being taken relative to $[c,d]$, and its derivative is $(x|_{[c,d]})'=x'|_{[c,d]}$. The same holds for a differentiable $A:[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$ in place of $x$.