# Definition of a set of class $C^1$ in $R^1$.

Main Question: What is the definition of a set with $$C^1$$ boundary when that set is a subset of $$\mathbb{R}^1$$?

In particular, given the definition of a set with $$C^1$$ boundary from page 710 of Evans PDE book (or the equivalent definition from the Brezis book) it is not clear to me what it means for $$\partial U$$ to be $$C^1$$ when $$U \subset \mathbb{R}^1$$. Is such a $$U$$ trivially $$C^1$$?

Bonus Question: If it is relevant to the answer, I am interested in verifying that the set $$U=(-1,1)^n$$ has $$C^1$$ boundary even when $$n=1$$ so I can apply the Sobolev embedding of Theorem 5 from Chapter 5.6.2 in Evans PDE (p283).

For $$n=1$$, we should consider $$\gamma$$ as a constant number, which then has to be equal to the boundary point. Thus, the definition reads: For each $$x^0 \in \partial U \subset \mathbb{R}$$ there should be a $$r>0$$ such that $$U \cap (x^0-r,x^0+r) = (x^0-r,x^0)$$ or $$U \cap (x^0-r,x^0+r) = (x^0,x^0+r)$$.
Clearly, $$(-1,1)$$ satisfies this with any $$r<1$$. However, a set such as $$(-1,1) \cup (1,2)$$ does not satisfy this.