Partial Derivatives, Continuity and $C^k$ Regularity under a Scaling Substitution

Let $n,m$ be natural numbers and let $\mathbb{R}$ be the real numbers, with absolute value $|\cdot|$. Regard Euclidean space $\mathbb{R}^n$ as a real vector space, and let $\lVert\,\cdot\,\rVert$ be the Euclidean norm and $d_E$ the Euclidean distance, a metric on $\mathbb{R}^n$.

Let $c\in\mathbb{R}^n$, let $\lambda$ and $\mu$ be nonzero real numbers, let $U\subseteq\mathbb{R}^n$ be open, and let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$. Put

$$V=\{x\in\mathbb{R}^n:\ c+\lambda x\in U\},$$

and let $g=(g_1,\dots,g_m):V\to\mathbb{R}^m$ be given by $g(x)=\mu\,f(c+\lambda x)$, so that $g_j(x)=\mu\,f_j(c+\lambda x)$ for every $j\in\{1,\dots,m\}$.

1. (Domain) The map $T:\mathbb{R}^n\to\mathbb{R}^n$ given by $T(x)=c+\lambda x$ is a bijection, with inverse $S(y)=\lambda^{-1}(y-c)$, and $T$ restricts to a bijection from $V$ onto $U$. The set $V$ is open in $\mathbb{R}^n$. Moreover

$$U=\{y\in\mathbb{R}^n:\ (-\lambda^{-1}c)+\lambda^{-1}y\in V\},\qquad f(y)=\mu^{-1}\,g\bigl((-\lambda^{-1}c)+\lambda^{-1}y\bigr)\ \ (y\in U);$$

that is, $(f,U)$ arises from $(g,V)$ by a substitution of the same form, with $c$, $\lambda$, $\mu$ replaced by $-\lambda^{-1}c$, $\lambda^{-1}$, $\mu^{-1}$.

2. (Partial derivatives) Let $x\in V$, put $a=c+\lambda x$, and let $i\in\{1,\dots,n\}$ and $j\in\{1,\dots,m\}$. Then the partial derivative of $g_j$ with respect to the $i$th variable exists at $x$ if and only if the partial derivative of $f_j$ with respect to the $i$th variable exists at $a$, and in that case

$$\partial_i g_j(x)=\mu\lambda\,\partial_i f_j(a).$$

3. (Continuity) For every $j\in\{1,\dots,m\}$: the function $f_j$ is continuous at every point of $U$ if and only if $g_j$ is continuous at every point of $V$.

4. (Class $C^k$) For every natural number $k$: $f$ is of class $C^k$ on $U$ if and only if $g$ is of class $C^k$ on $V$.

5. (Smoothness) $f$ is smooth on $U$ if and only if $g$ is smooth on $V$.