Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set

Let $n$ be a natural number, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, let $f,g:U\to\mathbb{R}$, and let $c\in\mathbb{R}$. Write $f+g$, $cf$ and $fg$ for the pointwise sum, scalar multiple and product on $U$, given by

$$(f+g)(x)=f(x)+g(x),\qquad (cf)(x)=c\,f(x),\qquad (fg)(x)=f(x)\,g(x)\qquad(x\in U).$$

1. (Partial derivatives) Let $x\in U$ and $i\in\{1,\dots,n\}$, and suppose that the partial derivatives of $f$ and of $g$ with respect to the $i$th variable exist at $x$. Then those of $f+g$, $cf$ and $fg$ exist at $x$, and

$$\partial_i(f+g)(x)=\partial_i f(x)+\partial_i g(x),\qquad \partial_i(cf)(x)=c\,\partial_i f(x),$$ $$\partial_i(fg)(x)=\partial_i f(x)\,g(x)+f(x)\,\partial_i g(x).$$

2. (Constants and coordinate functions) For every $b\in\mathbb{R}$, the function $U\to\mathbb{R}$ with constant value $b$ is smooth on $U$. For every $l\in\{1,\dots,n\}$, the $l$th coordinate function $\pi_l:U\to\mathbb{R}$ given by $\pi_l(x)=x_l$ is smooth on $U$.

3. (Sums, scalar multiples and products) Let $k$ be a natural number. If $f$ and $g$ are of class $C^k$ on $U$, then $f+g$, $cf$ and $fg$ are of class $C^k$ on $U$. If $f$ and $g$ are smooth on $U$, then $f+g$, $cf$ and $fg$ are smooth on $U$.