Characterization of $C^\alpha$-differentiability on a submanifold with boundary

Question

Let $d\in\mathbb N$, $M\subseteq\mathbb R^d$, $x\in M$, $E$ be a $\mathbb R$-Banach space, $f:M\to E$ and $\alpha\in\mathbb N_0\cup\{\infty\}$.

I was able to show the following:

1. If $$f\circ\phi^{-1}\in C^\alpha(\phi(\Omega),E)\tag1$$ for some $k$-dimensional interior $C^\alpha$-chart$^2$ $(\Omega,\phi)$ of $M$ for some $k\in\{1,\ldots,d\}$, then $f$ is $C^\alpha$-differentiable$^1$ at $x$.
2. If $f$ is $C^\alpha$-differentiable at $x$ and $M$ is a $k$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ without boundary for some $k\in\{1,\ldots,d\}$, then there is a $k$-dimensional interior $C^\alpha$-chart $(\Omega,\phi)$ with $(1)$.

Question: Are we able to show an analogous equivalent characterization of $C^\alpha$-differentiability in the case where $M$ is a $k$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary for some $k\in\{1,\ldots,d\}$?

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$^1$ If $E_i$ is a $\mathbb R$-Banach space and $\Omega_i\subseteq E_i$, then $f:\Omega_1\to E_1$ is called $C^\alpha$-differentiable at $x_1\in\Omega_1$ if $$\left.f\right|_{O_1\:\cap\:\Omega_1}=\left.\tilde f\right|_{O_1\:\cap\:\Omega_1}$$ for some $\tilde f\in C^\alpha(O_1,E_2)$ for some $E_1$-open neighborhood $O_1$ of $x_1$. $f$ is called $C^\alpha$-differentiable if $f$ is $C^\alpha$ differentiable at $x_1$ for all $x_1\in\Omega_1$. If $\Omega_2\subseteq E_2$, then $f$ is called $C^\alpha$-diffeomorphism from $\Omega_1$ onto $\Omega_2$ if $f$ is a homeomorphism from $\Omega_1$ onto $\Omega_2$ and $f$ and $f^{-1}$ are $C^\alpha$-differentiable.

$^2$ $(\Omega,\phi)$ is called $k$-dimensional $C^\alpha$-chart of $M$ if $\Omega$ is an open subset of $M$ and $\phi$ is a $C^\alpha$-diffeomorphism from $\Omega$ onto an open subset of $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. In that case, $(\Omega,\phi)$ is called $k$-dimensional interior $C^\alpha$-chart of $M$ if $\phi(\Omega)$ is $\mathbb R^k$-open.