What discuss below is a reference form the text "Analysis on Manifolds" by James Munkres.
First of all we observe the the interior of $U$ is not empty when $U$ is too: for details see here.
So by the first definition to we can suppose that the derivative of $\alpha$ exist only in $\text{int}(U)$ but strangerly Munkres state that $D\beta(x)=D\alpha(x)$ for any $x\in U$. So how explain this statement?
Could it be a typo? So could someone help me, please?
Best Answer
I'm not sure I properly understand what your problem is, but Munkres nowhere explicitly writes down the equation $\ D\alpha(x)=D\beta(x)\ $, or implies that $\ D\alpha(x)\ $ exists at a boundary point of $\ U\ $ in any sense other than what he is here showing can be taken to be its definition there. The point of his Lemma $23.2$ is that if $\ \beta_1\ $ and $\ \beta_2\ $ are any $\ C^r\ $ extensions of $\ \alpha\ $ to an open subset of $\ \mathbf{R}^k\ $, then $\ D\beta_1(x)= D\beta_2(x)\ $ for all $\ x\in U\ $. Thus, even though the normal definition of $\ D\alpha(x)\ $ cannot be used to define it on the intersection of $\ U\ $ with the boundary of $\ \mathbf{H}^k\ $ , you can nevertheless take its definition to be $\ D\beta(x)\ $ there, where $\ \beta\ $ is any extension of $\ \alpha\ $ to an open subset of $\ \mathbf{R}^k\ $.