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

Throughout, for $x\in U$, an index $i\in\{1,\dots,n\}$ and a real number $s$, write $x[s]$ for the point of $\mathbb{R}^n$ whose $i$th coordinate is $s$ and whose $k$th coordinate is $x_k$ for every $k\ne i$. By claim 1 of Slice Function and the Partial Derivative there is a real $\varrho>0$ such that $x[s]\in U$ whenever $|s-x_i|<\varrho$; this condition involves only $U$, so one such $\varrho$ serves for every function on $U$ at once. Let $I=\{s\in\mathbb{R}:|s-x_i|<\varrho\}$, an open interval all of whose points are interior by An Open Interval is an Interval All of Whose Points Are Interior; in particular $x_i$ is an interior point of $I$. For a function $u:U\to\mathbb{R}$ write $\tilde u:I\to\mathbb{R}$, $\tilde u(s)=u(x[s])$, for its slice function at $x$ in the $i$th variable as in claim 2 of Slice Function and the Partial Derivative.

We also record the following, used repeatedly. Let $d_E$ be the Euclidean distance, a metric on $\mathbb{R}^n$, and $d_{\mathbb{R}}$ the metric of The Absolute Value Metric on the Real Numbers on $\mathbb{R}$. By claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, a function $u:U\to\mathbb{R}$ is continuous at a point of $U$ if and only if it is continuous there relative to $U$ as a map into $(\mathbb{R},d_{\mathbb{R}})$; consequently, by claims 1 to 4 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, constant functions are continuous at every point of $U$ and sums, scalar multiples and products of functions continuous at every point of $U$ are continuous at every point of $U$. We refer to this as the continuity toolkit.

Claim 1. Directly from the definitions, the slice functions at $x$ in the $i$th variable of $f+g$, $cf$ and $fg$ are the pointwise sum $\tilde f+\tilde g$, scalar multiple $c\tilde f$ and product $\tilde f\tilde g$ on $I$. By claim 2 of Slice Function and the Partial Derivative, $\tilde f$ and $\tilde g$ are differentiable at $x_i$ with $\tilde f'(x_i)=\partial_i f(x)$ and $\tilde g'(x_i)=\partial_i g(x)$. By claims 2 and 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\tilde f+\tilde g$, $c\tilde f$ and $\tilde f\tilde g$ are differentiable at $x_i$ with derivatives $\tilde f'(x_i)+\tilde g'(x_i)$, $c\,\tilde f'(x_i)$ and $\tilde f'(x_i)\tilde g(x_i)+\tilde f(x_i)\tilde g'(x_i)$. Applying claim 2 of Slice Function and the Partial Derivative in the other direction to $f+g$, $cf$ and $fg$, their partial derivatives with respect to the $i$th variable exist at $x$ and equal those numbers; since $\tilde f(x_i)=f(x)$ and $\tilde g(x_i)=g(x)$, these are the asserted values.

Claim 2. Constants. For $b\in\mathbb{R}$ let $\kappa_b:U\to\mathbb{R}$ have constant value $b$. It is continuous at every point of $U$ by the continuity toolkit. For every $x\in U$ and every $i$, the slice function of $\kappa_b$ at $x$ in the $i$th variable is the function on $I$ with constant value $b$, which is differentiable at $x_i$ with derivative $0$ by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives; so by claim 2 of Slice Function and the Partial Derivative the partial derivative of $\kappa_b$ with respect to the $i$th variable exists at every point of $U$, and $\partial_i\kappa_b=\kappa_0$.

We show by induction on $k$ that every constant function on $U$ is of class $C^k$ on $U$. For $k=1$: $\kappa_b$ is continuous at every point of $U$ and, for each $i$, $\partial_i\kappa_b$ exists at every point of $U$ and the function $\partial_i\kappa_b=\kappa_0$ is continuous at every point of $U$; so $\kappa_b$ is of class $C^1$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set. If every constant function on $U$ is of class $C^k$, then $\kappa_b$ is of class $C^1$ and each $\partial_i\kappa_b=\kappa_0$ is of class $C^k$, so $\kappa_b$ is of class $C^{k+1}$ by clauses 2 and 3 of C^k Maps on a Euclidean Open Set. Hence $\kappa_b$ is of class $C^k$ for every natural number $k$, that is, smooth on $U$ by Smooth Map on a Euclidean Open Set.

Coordinate functions. Fix $l$. For $x,a\in U$ we have $|\pi_l(x)-\pi_l(a)|=|x_l-a_l|\le\lVert x-a\rVert=d_E(x,a)$ by claims 4 and 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $\pi_l$ is continuous at every point of $U$ relative to $U$ as a map into $(\mathbb{R},d_{\mathbb{R}})$ (given $\varepsilon>0$ take the threshold $\varepsilon$), hence continuous at every point of $U$.

Fix $x\in U$ and $i$. If $l\ne i$ then $(x[s])_l=x_l$ for every $s$, so the slice function of $\pi_l$ at $x$ in the $i$th variable is constant, with derivative $0$ at $x_i$ by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives. If $l=i$ then $(x[s])_l=s$, so the slice function is $s\mapsto s$ on $I$; for every real $\varepsilon>0$ and every real $h$ with $0<|h|<\varrho$ and $x_i+h\in I$ one has $\bigl((x_i+h)-x_i\bigr)/h=1$, so by Derivative at an Interior Point this slice function is differentiable at $x_i$ with derivative $1$. In both cases claim 2 of Slice Function and the Partial Derivative shows that the partial derivative of $\pi_l$ with respect to the $i$th variable exists at every point of $U$, and $\partial_i\pi_l$ is the constant function with value $1$ if $l=i$ and with value $0$ otherwise.

Each $\partial_i\pi_l$ is therefore a constant function, hence smooth on $U$ by the previous paragraph and so of class $C^k$ on $U$ for every natural number $k$; in particular it is of class $C^1$, hence continuous at every point of $U$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set. Thus $\pi_l$ is of class $C^1$ on $U$, and for every natural number $k$ the function $\pi_l$ is of class $C^1$ with every $\partial_i\pi_l$ of class $C^k$, so $\pi_l$ is of class $C^{k+1}$ by clause 2 of C^k Maps on a Euclidean Open Set. Hence $\pi_l$ is of class $C^k$ on $U$ for every natural number $k$, that is, smooth on $U$.

Claim 3. We argue by induction on $k$, the assertion being proved simultaneously for all $f$, $g$ and $c$ as in the statement.

Base case $k=1$. Suppose $f$ and $g$ are of class $C^1$ on $U$. By clauses 1 and 3 of C^k Maps on a Euclidean Open Set, $f$ and $g$ are continuous at every point of $U$ and, for each $i$, the partial derivatives of $f$ and $g$ with respect to the $i$th variable exist at every point of $U$ and the functions $\partial_i f,\partial_i g:U\to\mathbb{R}$ are continuous at every point of $U$. By the continuity toolkit, $f+g$, $cf$ and $fg$ are continuous at every point of $U$. By claim 1 their partial derivatives with respect to the $i$th variable exist at every point of $U$, and as functions on $U$,

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

each of which is continuous at every point of $U$ by the continuity toolkit. Hence $f+g$, $cf$ and $fg$ are of class $C^1$ on $U$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set.

Induction step. Let $k$ be a natural number, assume the assertion for $k$, and let $f$ and $g$ be of class $C^{k+1}$ on $U$. By clause 2 of C^k Maps on a Euclidean Open Set both are of class $C^1$ on $U$ and, for each $i$, $\partial_i f$ and $\partial_i g$ are of class $C^k$ on $U$; by claim 2 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map both $f$ and $g$ are also of class $C^k$ on $U$. By the base case, $f+g$, $cf$ and $fg$ are of class $C^1$ on $U$, with the partial derivatives displayed above. The induction hypothesis, applied to the pairs of $C^k$ functions occurring there, gives that $\partial_i f+\partial_i g$, $c\,\partial_i f$, $(\partial_i f)g$, $f(\partial_i g)$ and finally $(\partial_i f)g+f(\partial_i g)$ are of class $C^k$ on $U$. So every first partial derivative of $f+g$, of $cf$ and of $fg$ is of class $C^k$ on $U$, and clauses 2 and 3 of C^k Maps on a Euclidean Open Set give that these three functions are of class $C^{k+1}$ on $U$.

Smooth case. If $f$ and $g$ are smooth on $U$ then by Smooth Map on a Euclidean Open Set they are of class $C^k$ on $U$ for every natural number $k$, so the same holds for $f+g$, $cf$ and $fg$, which are therefore smooth on $U$. $\blacksquare$