Smoothness of functions up to the boundary Let $\Omega$ be an open set in $R^n$ with $C^1$ boundary $\partial \Omega$.
Suppose that $f$ is a $C^1$ function on $\Omega \cup \partial \Omega $, where $C^1$-smoothness of $f(x)$ on  $\partial \Omega$ is defined via local $C^1$- diffeomorphism.
If we define a function $g$ on $\Omega$ as $g(x) = f(x)$ for $x \in \Omega$, and fix $x_0 \in \partial \Omega$, then is it true that $D^{\alpha} g(x) \to D^{\alpha} f(x_0)$ as $x \to x_0$ for each multiindex $|\alpha| \leq 1$?

No, it's not true, because your definition of being $C^1$ on $\Omega 
\cup \partial \Omega$ is bad.
Your definition only guarantees that the restriction of $f$ to the boundary is $C^1$, but that is not good enough: $f$ can be $C^1$ on $\Omega$ and on $\partial \Omega$ without being even continuous on $\Omega 
\cup \partial \Omega$.
You should say that $f$ is $C^1$ at points of $\partial \Omega$ if it admits a $C^1$ extension in small open sets around them. This is a good definition and it does not require $\partial \Omega$ to have any kind of regularity.
