Restriction of a $C^k$ Map to an Open Subset

Let $n,m$ be natural numbers and let $\mathbb{R}$ be the real numbers. Let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$ and let $V\subseteq U$ be open in $\mathbb{R}^{n}$.

Let $f=(f_{1},\dots,f_{m}):U\to\mathbb{R}^{m}$ and $g:U\to\mathbb{R}$, and let $f|_{V}:V\to\mathbb{R}^{m}$ and $g|_{V}:V\to\mathbb{R}$ be their restrictions to $V$, whose values at $x\in V$ are $f(x)$ and $g(x)$; the coordinate functions of $f|_{V}$ are the restrictions $f_{j}|_{V}$.

Let $i\in\{1,\dots,n\}$, let $x\in V$ and let $L\in\mathbb{R}$. Then the following hold.

1. (Partial derivatives) The partial derivative of $g|_{V}$ with respect to the $i$th variable exists at $x$ with value $L$ if and only if the partial derivative of $g$ with respect to the $i$th variable exists at $x$ with value $L$. In particular one exists at $x$ exactly when the other does, with the same admissible values, and in that case we write $\partial_{i}(g|_{V})(x)=\partial_{i}g(x)$.

2. (Continuity) If $f$ is continuous at $x$, then $f|_{V}$ is continuous at $x$; the same holds with $g$ and $g|_{V}$ in place of $f$ and $f|_{V}$.

3. (Class $C^{k}$) For every natural number $k$: if $f$ is of class $C^{k}$ on $U$, then $f|_{V}$ is of class $C^{k}$ on $V$. By the scalar convention of clause 3 of that definition, the same applies to $g$ and $g|_{V}$.

4. (Smoothness) If $f$ is smooth on $U$, then $f|_{V}$ is smooth on $V$; likewise for $g$ and $g|_{V}$.