Short question: How is $C^1(K)$ defined for a compact set $K$?

Question

Let $K\subset\mathbb R^d$ be compact. Then how is $C^1(K)$ defined? I've seen this at many places, but no one seems to care about the problems at the boundary of $K$.

Answer 1

It probably means "the set of restrictions of functions that are defined on an open set containing $K$, and that are $C^1$ on the open set".

I think I remember from differential geometry courses that whenever $K$ is a submanifold of $\mathbb{R}^n$, the intrinsic definition of $C^1(K)$ coincides with this one, therefore it is a "good definition".

Note that any $C^1$ function in the above sense is obviously $C^1$ in @Surb's sense. For $d = 1$, the definitions agree: let $f: [a,b] \rightarrow \mathbb{R}$ that is continuous, $C^1$ on the interior of the interval, and such that $j(a) := \lim_{x\to a} f'(x)$ and $j(b) :=\lim_{x\to b} f'(x)$ exist. Then define $g$ as $f$ on $[a,b]$ and, on $(a-1,a)$ to be $t \mapsto (t-a)j(a) + f(a)$ and on $(b, b+1)$ to be $t \mapsto (t-b)j(b) + f(b)$. Then it is easy to show that $g$ is $C^1$ on $(a-1,b+1)$. I don't know if the same is true in higher dimensions.

Answer 2

I guess that the problem comes from the derivative on the boundary of $K$. Suppose that $\text{int}(K)$ is a domain. Normally, $f\in \mathcal C^1(K)$ means that $f\in \mathcal C^0(K)\cap \mathcal C^{1}(\text{int}(K))$ s.t. for all $u\in Bd(K)$, the limit $$\lim_{\substack{x\to u \\ x\in \text{int}(K)}}f'(x)$$ exist. I let you determine what means $f'$ in this context (it's the gradient if $f:K\to \mathbb R$, or the Jacobian if $f:K\to \mathbb R^m$ for some $m\geq 2$).

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So, for $d=1$, then $K=[a,b]$ for some $a,b\in \mathbb R$, $a<b$ and $f:K\to \mathbb R$. So, namely $f\in \mathcal C^1([a,b])$ means that $f\in \mathcal C^0([a,b])\cap \mathcal C^1((a,b))$ s.t. $$\lim_{x\to a^+}f'(x)\quad \text{and}\quad \lim_{x\to b^-}f'(x),$$ exists.

Answer 3

$$C^m(\bar{\Omega}) := \{ f \in C^m(\Omega); \: \forall |\alpha| \leq m, \exists \: g^{\alpha} \in C({\bar{\Omega}}) \: s.t. \: \partial^{\alpha}f = g^{\alpha}|_{\Omega} \}$$

ln other words, $C^m(\bar{\Omega})$ consists of all functions $f \in C^m(\Omega)$ that, together with all their partial derivatives $\partial^{\alpha}f \:, 1 \leq \alpha \leq m $ possess continuous extensions to $\bar{\Omega}$ or equivalently, such that, at each point $x_0 \in \partial \Omega, \lim_{x \rightarrow x_0} \partial^{\alpha}f(x)$ exists in $\mathbb{R} \:, \: 0 \leq \alpha \leq m $ or equivalently, when $\Omega$ is bounded, if each function $\partial^{\alpha} f, \: 0 \leq \alpha \leq m$ is uniformly continuous in $\Omega$.
Source: Linear and Nonlinear Functional Analysis (section 1.18) by Philippe G. Ciarlet