Proof of Restriction of a Cᵏ Function to an Open Subset

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) and Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ix), and norms and distances in $\mathbb{R}^{n}$ through Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v) and Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length (iii); moreover $f|_{W}$ has the values of $f$ on $W$ (Domains and Restrictions of Maps (iii)), so that the component maps of $f|_{W}$ are the restrictions $(f_{j})|_{W}$; and $\partial_{i}$, $\nabla$ and $D$ for functions on $W$ are as in Higher Derivatives on Open Subsets of ℝⁿ: Notation (iii).

We begin with (i). Let $x\in W$ and $i\in[n]$. Since $W$ is open (Open Subset of a Metric Space (i)), there is a real $r>0$ with $B(x,r)\subseteq W$. Let $f:U\to\mathbb{R}$ and $L\in\mathbb{R}$, and suppose first that $L$ is a partial derivative of $f$ with respect to the $i$-th variable at $x$ in the sense of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (i). Given a real $\varepsilon>0$, choose $\delta>0$ for $\varepsilon$ as in that definition, and put $\delta'=\min(\delta,r)$. Every real $t$ with $0<|t|<\delta'$ satisfies

$$d_{E}(x+te_{i},x)=\lVert te_{i}\rVert=|t|<r,$$

so $x+te_{i}\in B(x,r)\subseteq W$; and since $f|_{W}$ agrees with $f$ at $x$ and at $x+te_{i}$,

$$\frac{f|_{W}(x+te_{i})-f|_{W}(x)}{t}=\frac{f(x+te_{i})-f(x)}{t},$$

which differs from $L$ by less than $\varepsilon$ because $|t|<\delta$. Thus $\delta'$ serves for $f|_{W}$, and $L$ is a partial derivative of $f|_{W}$ at $x$. Conversely, if $\delta>0$ serves for $f|_{W}$ and $\varepsilon$, then every $t$ with $0<|t|<\delta$ has $x+te_{i}\in W\subseteq U$, and the same identity of difference quotients shows that $\delta$ serves for $f$. Hence $L$ is a partial derivative of $f|_{W}$ at $x$ exactly when it is one of $f$; in particular $\partial_{i}(f|_{W})(x)$ exists exactly when $\partial_{i}f(x)$ exists, and then the two are equal (Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (ii)).

The same argument applies (in both directions) to derivatives, with vectors $h\in\mathbb{R}^{n}$ satisfying $\lVert h\rVert<\min(\delta,r)$ in place of the reals $t$ with $0<|t|<\min(\delta,r)$: such an $h$ has $x+h\in B(x,r)\subseteq W$, and the remainders

$$f|_{W}(x+h)-f|_{W}(x)-w\cdot h=f(x+h)-f(x)-w\cdot h$$

agree, and likewise with $Ah$ in place of $w\cdot h$. So for every $w\in\mathbb{R}^{n}$, a function $f:U\to\mathbb{R}$ has $f|_{W}$ differentiable at $x$ with derivative vector $w$ exactly when $f$ is differentiable at $x$ with derivative vector $w$ (Differentiability at a Point for Functions Between Euclidean Spaces (ii)), and for every $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ the same holds for $f:U\to\mathbb{R}^{m}$ with derivative matrix $A$ (Differentiability at a Point for Functions Between Euclidean Spaces (i)). In particular $f|_{W}$ is differentiable at $x$ exactly when $f$ is. In that case $w=\nabla f(x)$ is a derivative vector of $f$ at $x$, hence of $f|_{W}$, and $A=Df(x)$ is a derivative matrix of $f$ at $x$, hence of $f|_{W}$; since the derivative vector is the gradient and the derivative matrix is the Jacobian matrix (The Derivative Is the Jacobian Matrix, and Is Unique (i), The Derivative Is the Jacobian Matrix, and Is Unique (ii)),

$$\nabla(f|_{W})(x)=\nabla f(x),\qquad D(f|_{W})(x)=Df(x).$$

We turn to (ii), first for real-valued functions. We show by induction on $k$ (The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (viii)) that every $h\in\mathcal{C}^{k}(U)$ has $h|_{W}\in\mathcal{C}^{k}(W)$, where the classes on $U$ and on $W$ are those of Functions of Class Cᵏ on an Open Subset of ℝⁿ (i). For $k=0$, the restriction of a function continuous on $U$ is continuous on $W$, by Restricting a Continuous Map to a Smaller Subset Preserves Continuity (ii). For the inductive step, let $h\in\mathcal{C}^{k+1}(U)$ and $i\in[n]$. By (i), $\partial_{i}(h|_{W})(x)$ exists for every $x\in W$ and equals $\partial_{i}h(x)$, that is,

$$\partial_{i}(h|_{W})=(\partial_{i}h)|_{W}.$$

Since $\partial_{i}h\in\mathcal{C}^{k}(U)$, the induction hypothesis gives $(\partial_{i}h)|_{W}\in\mathcal{C}^{k}(W)$. As this holds for every $i\in[n]$, $h|_{W}\in\mathcal{C}^{k+1}(W)$.

Finally, let $f:U\to\mathbb{R}^{m}$ be of class $C^{k}$ on $U$. Each component map $f_{j}$ lies in $\mathcal{C}^{k}(U)$, so by the real-valued case each component $(f_{j})|_{W}$ of $f|_{W}$ lies in $\mathcal{C}^{k}(W)$, and $f|_{W}$ is of class $C^{k}$ on $W$ (Functions of Class Cᵏ on an Open Subset of ℝⁿ (iii)). If $f$, real- or vector-valued, is smooth on $U$, it is of class $C^{k}$ on $U$ for every $k$, hence $f|_{W}$ is of class $C^{k}$ on $W$ for every $k$, that is, smooth on $W$ (Smooth Functions on an Open Subset of ℝⁿ (i), Smooth Functions on an Open Subset of ℝⁿ (ii)).