Proof of Basic Properties of Cᵏ Functions

Throughout, membership in $\mathcal{C}^{0}(U)$ and in $\mathcal{C}^{k+1}(U)$ is read through Functions of Class Cᵏ on an Open Subset of ℝⁿ (i), and continuity on $U$ means continuity at every point of $U$, by Continuous Map Between Metric Spaces (ii).

We begin with (i). For $f:U\to\mathbb{R}$ this is the base of the recursion: $\mathcal{C}^{0}(U)$ consists exactly of the functions continuous on $U$ (Functions of Class Cᵏ on an Open Subset of ℝⁿ (i), Functions of Class Cᵏ on an Open Subset of ℝⁿ (ii)). For $f:U\to\mathbb{R}^{m}$, $f$ is of class $C^{0}$ exactly when every component map $f_{j}$ is continuous on $U$ (Functions of Class Cᵏ on an Open Subset of ℝⁿ (iii)), and this holds exactly when $f$ is continuous on $U$, by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (i) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v).

Next, the real-valued part of (v). Let $f\in\mathcal{C}^{1}(U)$. Then for every $i\in[n]$ the partial derivative $\partial_{i}f(x)$ exists for every $x\in U$, and $\partial_{i}f\in\mathcal{C}^{0}(U)$, that is, $\partial_{i}f$ is continuous on $U$. By Continuous Partial Derivatives Imply Differentiability (i), applied at each $a\in U$, $f$ is differentiable at $a$ with derivative vector $\nabla f(a)$. The gradient function $\nabla f:U\to\mathbb{R}^{n}$ (Gradient of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (ii)) has the $\partial_{i}f$ as its component maps (Gradient of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (i)); since these are continuous on $U$, so is $\nabla f$, by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (i) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v).

For the vector-valued part of (v), let $f:U\to\mathbb{R}^{m}$ be of class $C^{1}$, so that every component map $f_{j}$ lies in $\mathcal{C}^{1}(U)$. As in the real-valued case, every partial derivative $\partial_{i}f_{j}$ exists on $U$ and is continuous on $U$; these are the hypotheses of Continuous Partial Derivatives Imply Differentiability (ii), which therefore shows that $f$ is differentiable at every $a\in U$ with derivative matrix $Df(a)$. The Jacobian function $Df:U\to\mathcal{M}_{m\times n}(\mathbb{R})$ has the entry maps

$$x\mapsto\bigl(Df(x)\bigr)_{ji}=\partial_{i}f_{j}(x)$$

(Jacobian Matrix of a Function Between Euclidean Spaces (i)), which are continuous on $U$; hence $Df$ is continuous on $U$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (ii) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v).

We turn to (ii), first for real-valued functions. We show

$$\mathcal{C}^{k+1}(U)\subseteq\mathcal{C}^{k}(U)\qquad\text{for every }k\in\mathbb{N}_{0}$$

by induction on $k$ (The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (viii)). For $k=0$, a function $f\in\mathcal{C}^{1}(U)$ is differentiable at every point of $U$ by the previous paragraphs, hence continuous at every point of $U$ by Differentiable Functions Are Continuous (i), so $f\in\mathcal{C}^{0}(U)$. For the inductive step, let $f\in\mathcal{C}^{k+2}(U)$. Every $\partial_{i}f$ exists on $U$ and lies in $\mathcal{C}^{k+1}(U)$, which is contained in $\mathcal{C}^{k}(U)$ by the induction hypothesis; so $f\in\mathcal{C}^{k+1}(U)$. Given $l\le k$, applying this inclusion $k-l$ times (an induction on $k-l$) gives $\mathcal{C}^{k}(U)\subseteq\mathcal{C}^{l}(U)$. A smooth $f$ lies in $\mathcal{C}^{k}(U)$ for every $k$ by Smooth Functions on an Open Subset of ℝⁿ (i).

The vector-valued case of (ii) follows by applying the real-valued case to every component map: a map $f:U\to\mathbb{R}^{m}$ is of class $C^{k}$, or smooth, exactly when each $f_{j}$ is, by Functions of Class Cᵏ on an Open Subset of ℝⁿ (iii) and Smooth Functions on an Open Subset of ℝⁿ (ii).

For (iii), let $f:U\to\mathbb{R}$ and $i\in[n]$. If $f\in\mathcal{C}^{k+1}(U)$, the conclusion is the defining condition of $\mathcal{C}^{k+1}(U)$ in Functions of Class Cᵏ on an Open Subset of ℝⁿ (i). If $f$ is smooth, then $f\in\mathcal{C}^{k+1}(U)$ for every $k$, so $\partial_{i}f$ exists on $U$ and lies in $\mathcal{C}^{k}(U)$ for every $k$; that is, $\partial_{i}f$ is smooth (Smooth Functions on an Open Subset of ℝⁿ (i)).

Finally, (iv): by (ii), a function of class $C^{k}$, or smooth, is of class $C^{0}$, hence continuous on $U$ by (i).