A description/proof of that \(C^k\)-ness of map from closed interval into subset of Euclidean \(C^\infty\) manifold at boundary point equals existence of one-sided derivatives with continuousness, and derivatives are one-sided derivatives
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous.
Target Context
- The reader will have a description and a proof of the proposition that \(C^k\)-ness of any map from any (possibly half) closed interval into any subset of any Euclidean \(C^\infty\) manifold at any closed boundary point equals the existence of the one-sided derivatives with continuousness, and the derivatives are the one-sided the derivatives, where \(k\) excludes \(0\) and \(\infty\).
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any Euclidean \(C^\infty\) manifolds, \(\mathbb{R}^d\) and \(\mathbb{R}\), any (possibly half) closed interval, \(J \subseteq \mathbb{R}\), with the lower and upper boundary points, \(t_1 \lt t_2\) (\(t_1 = t_2\) is excluded because for \([t_1, t_1]\), one-sided derivatives do not make sense), any subset, \(S \subseteq \mathbb{R}^d\), and any map, \(f: J \to S\), for any closed boundary point, \(t_j \in J\), \(f\) is \(C^k\) at \(t_j\) if and only if \(f\) has the one-side derivatives at \(t_j\) and the derivatives maps are continuous over an open neighborhood of \(t_j\) on \(J\), and the derivatives are the one-sided derivatives.
2: Proof
Let us suppose that \(f\) is \(C^k\) at \(t_j\).
There are an open neighborhood, \(U'_{t_j} = (t_j - \epsilon, t_j + \epsilon) \subseteq \mathbb{R}\), of \(t_j\) and a \(C^k\) map, \(f': U'_{t_j} \to \mathbb{R}^d\), such that \(f' \vert_{U'_{t_j} \cap J} = f \vert_{U'_{t_j} \cap J}\), by the definition.
\(d^k f / d t^k \vert_{t_j}\) is defined to be \(d^k f' / d t^k \vert_{t_j}\), but \(d^k f' / d t^k \vert_{t_j}\) equals the one-sided derivative of \(f\), so, the one-sided derivative exists; besides, \(f\) has the derivatives over open \(U'_{t_j} \cap (J \setminus \{t_j\})\), because \(f' = f\) there and \(f'\) has the derivatives there, and as \(d^k f' / d t^k: U'_{t_j} \cap J \to \mathbb{R}^d\) is continuous, \(d^k f / d t^k: U'_{t_j} \cap J \to \mathbb{R}^d\) (the value at \(t_j\) is the one-sided one) is continuous.
Let us suppose that \(f\) has the one-side derivatives at \(t_j\) and there is an open neighborhood, \(U'_{t_j} = (t_j - \epsilon, t_j + \epsilon) \subseteq \mathbb{R}\), of \(t_j\) and \(d^k f / d t^k: U'_{t_j} \cap J \to \mathbb{R}^d\) (the value at \(t_j\) is the one-sided one) is continuous.
Let us suppose that \(t_j\) is the upper boundary point, \(t_j = t_2\).
\(\epsilon\) can be taken to be \(t_1 \lt t_j - \epsilon\).
Denoting the \(l\)-th one-sided derivative as \(a_l\), let us define \(f': U'_{t_j} \to \mathbb{R}^d\) as \(f' = f\) over \(U'_{t_j} \cap J\) and \(f' = f (t_1) + a_1 (t - t_j) + 1 / 2 a_2 (t - t_j)^2 + ... + 1 / k! a_k (t - t_j)^k\) elsewhere. \(f'\) has the derivatives on the whole \(U'_{t_j}\), and the derivatives maps are continuous over \(U'_{t_j}\), because the derivatives equal \(d^l f / d t^l\) over \(U'_{t_j} \cap J = (t_j - \epsilon, t_j]\), continuous over the closed subspace of \(U'_{t_j}\), and equal \(a_l\) over \((U'_{t_j} \setminus (U'_{t_j} \cap J)) \cup \{t_j\} = [t_j, t_j + \epsilon)\), continuous over the closed subspace of \(U'_{t_j}\), by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous.
So, \(f\) is \(C^k\) at \(t_j\).
Anyway, the derivatives at the boundary point are the one-sided derivatives.
3: Note
\(k = \infty\) is excluded, because the corresponding infinite series, \(f' = f (t_1) + a_1 (t - t_j) + 1 / 2 a_2 (t - t_j)^2 + ...\), has not been proved to converge.
But the former direction holds also for the \(k = \infty\) case.
