Proof of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions

Throughout, the arithmetic and order of $\mathbb{R}$ are used through The Structure of ℝ as a Field and an Ordered Ring (i), The Structure of ℝ as a Field and an Ordered Ring (ii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ix), Rules of Arithmetic and Order in an Ordered Field (iii) and The Quotient 1/x, Order Rules for Sums, and Nesting of Closed Intervals in ℝ (i), and the metric spaces $(\mathbb{R}^{n},d_{E})$ and $(\mathbb{R},d_{\mathbb{R}})$ through $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (i) and The Absolute Value Metric on the Real Numbers (i); membership in $\mathcal{C}^{0}(U)$ and $\mathcal{C}^{k+1}(U)$ is read through Functions of Class Cᵏ on an Open Subset of ℝⁿ (i), and each induction on $k$ runs from $0$ over $\mathbb{N}_{0}$ (The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (viii)), $k+1$ being the successor of $k$. Two reductions are used for every clause. A vector-valued statement follows from the real-valued one applied to the component maps (Functions of Class Cᵏ on an Open Subset of ℝⁿ (iii)), since the components of $f+g$ and $\lambda f$ are $f_{j}+g_{j}$ and $\lambda f_{j}$, and those of a constant function are constant. A statement about smooth functions follows from the $C^{k}$ statement for every $k$ (Smooth Functions on an Open Subset of ℝⁿ (i), Smooth Functions on an Open Subset of ℝⁿ (ii)).

We first compute the first partial derivatives of sums, scalar multiples, products and reciprocals. Let $f,g\in\mathcal{C}^{1}(U)$, $\lambda\in\mathbb{R}$ and $i\in[n]$. By Basic Properties of Cᵏ Functions (v), $f$ and $g$ are differentiable at every $x\in U$, with derivative vectors $\nabla f(x)$ and $\nabla g(x)$. By Rules for Derivatives of Functions on Open Subsets of ℝⁿ (v), Rules for Derivatives of Functions on Open Subsets of ℝⁿ (vi) and, when $g(x)\neq0$ for every $x\in U$, Rules for Derivatives of Functions on Open Subsets of ℝⁿ (vii), the functions $f+g$, $\lambda f$, $fg$ and $1/g$ are differentiable at every $x\in U$, with derivative vectors $\nabla f(x)+\nabla g(x)$, $\lambda\nabla f(x)$, $f(x)\nabla g(x)+g(x)\nabla f(x)$ and $-\frac{1}{g(x)^{2}}\nabla g(x)$. The $i$-th component of the derivative vector of a differentiable function is its $i$-th partial derivative (The Derivative Is the Jacobian Matrix, and Is Unique (i)); reading off $i$-th components, the partial derivatives below exist at every point of $U$, and, as functions on $U$ with the pointwise operations,

$$\partial_{i}(f+g)=\partial_{i}f+\partial_{i}g,\qquad\partial_{i}(\lambda f)=\lambda\,\partial_{i}f,\qquad\partial_{i}(fg)=f\,\partial_{i}g+g\,\partial_{i}f,$$ $$\partial_{i}\Bigl(\frac{1}{g}\Bigr)=(-1)\Bigl(\frac{1}{g}\cdot\frac{1}{g}\Bigr)\partial_{i}g\qquad\text{if }g(x)\neq0\text{ for every }x\in U.$$

We begin with (i), showing by induction on $k$ that every constant function $U\to\mathbb{R}$ lies in $\mathcal{C}^{k}(U)$. For $k=0$, constant functions are continuous on $U$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (i). For the inductive step, let $\varphi$ be constant. By Rules for Derivatives of Functions on Open Subsets of ℝⁿ (i), $\varphi$ is differentiable at every $x\in U$ with $\nabla\varphi(x)=0$, so $\partial_{i}\varphi(x)=0$ for every $x\in U$ and $i\in[n]$ (The Derivative Is the Jacobian Matrix, and Is Unique (i)). Thus each $\partial_{i}\varphi$ is the constant function $0$, which lies in $\mathcal{C}^{k}(U)$ by the induction hypothesis, and $\varphi\in\mathcal{C}^{k+1}(U)$.

For (ii), let $\varphi(x)=v\cdot x+c$. By Rules for Derivatives of Functions on Open Subsets of ℝⁿ (ii), $\varphi$ is differentiable at every point of $U$ with $\nabla\varphi(x)=v$; it is therefore continuous on $U$ by Differentiable Functions Are Continuous (i), so $\varphi\in\mathcal{C}^{0}(U)$. Its partial derivatives $\partial_{i}\varphi$ are the constant functions with value $v_{i}$ (The Derivative Is the Jacobian Matrix, and Is Unique (i)), which lie in $\mathcal{C}^{k}(U)$ for every $k$ by (i); hence $\varphi\in\mathcal{C}^{k+1}(U)$ for every $k$, and $\varphi$ is smooth. For $f(x)=Bx+c$, let $b_{j}=(B_{j1},\dots,B_{jn})\in\mathbb{R}^{n}$ be the $j$-th row of $B$. By Matrix-Vector Product (i),

$$f_{j}(x)=(Bx)_{j}+c_{j}=\sum_{l=1}^{n}B_{jl}x_{l}+c_{j}=b_{j}\cdot x+c_{j},$$

the last equality by The Dot Product on $\mathbb{R}^n$ (i); so every component of $f$ is smooth by the real-valued case, and $f$ is smooth.

For (iii), $\pi_{i}(x)=x_{i}=e_{i}\cdot x+0$, since $e_{i}\cdot x=x_{i}$ by Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length (i), so $\pi_{i}$ is smooth by (ii).

For (iv), we show by induction on $k$ that $f,g\in\mathcal{C}^{k}(U)$ and $\lambda\in\mathbb{R}$ imply $f+g,\lambda f\in\mathcal{C}^{k}(U)$. For $k=0$ this is the continuity of sums and scalar multiples of continuous functions, Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v). For the inductive step, let $f,g\in\mathcal{C}^{k+1}(U)$ and $i\in[n]$. By Basic Properties of Cᵏ Functions (ii), $f,g\in\mathcal{C}^{1}(U)$, so the first paragraph gives $\partial_{i}(f+g)=\partial_{i}f+\partial_{i}g$ and $\partial_{i}(\lambda f)=\lambda\,\partial_{i}f$. By Basic Properties of Cᵏ Functions (iii), $\partial_{i}f,\partial_{i}g\in\mathcal{C}^{k}(U)$, and the induction hypothesis gives $\partial_{i}(f+g),\partial_{i}(\lambda f)\in\mathcal{C}^{k}(U)$. As $i$ was arbitrary, $f+g,\lambda f\in\mathcal{C}^{k+1}(U)$.

For (v), we show by induction on $k$ that $f,g\in\mathcal{C}^{k}(U)$ imply $fg\in\mathcal{C}^{k}(U)$. For $k=0$ this is the continuity of products of continuous functions, Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v). For the inductive step, let $f,g\in\mathcal{C}^{k+1}(U)$ and $i\in[n]$. Since $f,g\in\mathcal{C}^{1}(U)$ by Basic Properties of Cᵏ Functions (ii), the first paragraph gives

$$\partial_{i}(fg)=f\,\partial_{i}g+g\,\partial_{i}f,$$

where $f,g\in\mathcal{C}^{k}(U)$ by Basic Properties of Cᵏ Functions (ii) and $\partial_{i}f,\partial_{i}g\in\mathcal{C}^{k}(U)$ by Basic Properties of Cᵏ Functions (iii). By the induction hypothesis both products on the right lie in $\mathcal{C}^{k}(U)$, and so does their sum, by (iv) at level $k$. Hence $fg\in\mathcal{C}^{k+1}(U)$.

For (vi), we show by induction on $k$ that every $g\in\mathcal{C}^{k}(U)$ with $g(x)\neq0$ for every $x\in U$ has $1/g\in\mathcal{C}^{k}(U)$. For $k=0$, let $E=\{t\in\mathbb{R}:t\neq0\}$. The identity map of $E$ is continuous on $E$ by Continuity of Identity Maps, Compositions and Reciprocals (i) and has no zeros, so the function $r:E\to\mathbb{R}$, $t\mapsto1/t$, is continuous on $E$ by Continuity of Identity Maps, Compositions and Reciprocals (iii). Since $g$ takes its values in $E$, $1/g=r\circ g$ is continuous on $U$ by Continuity of Identity Maps, Compositions and Reciprocals (ii). For the inductive step, let $g\in\mathcal{C}^{k+1}(U)$ have no zeros, and let $i\in[n]$. Since $g\in\mathcal{C}^{1}(U)$ by Basic Properties of Cᵏ Functions (ii), the first paragraph gives

$$\partial_{i}\Bigl(\frac{1}{g}\Bigr)=(-1)\Bigl(\frac{1}{g}\cdot\frac{1}{g}\Bigr)\partial_{i}g .$$

Here $g\in\mathcal{C}^{k}(U)$ by Basic Properties of Cᵏ Functions (ii), so $1/g\in\mathcal{C}^{k}(U)$ by the induction hypothesis, and $\partial_{i}g\in\mathcal{C}^{k}(U)$ by Basic Properties of Cᵏ Functions (iii). Two applications of (v) at level $k$ and one of (iv) with $\lambda=-1$ give $\partial_{i}(1/g)\in\mathcal{C}^{k}(U)$, so $1/g\in\mathcal{C}^{k+1}(U)$.

Finally, for (vii), $f/g=f\cdot(1/g)$ (Functions of Class Cᵏ: Notation (v)), so $f/g$ is of class $C^{k}$, respectively smooth, by (vi) and (v).