In Lee's book Introduction to Smooth Manifolds, he remarks on page 25 that
It is a fact (which we will neither prove nor use) that $F: U\rightarrow \mathbb R^k$ is smooth in this sense if and only if $F$ is continuous, $F|_{U\cap \text{Int }\mathbb H^n}$ is smooth, and the partial derivatives of $F|_{U\cap \text{Int }\mathbb H^n}$ of all orders have continuous extensions to all of $U$.
This follows a discussion of the definition of a smooth function on an arbitrary subset of $\mathbb R^n$. A function on such a set is defined to be smooth if and only if it admits a smooth extension to an open neighborhood of each point. The quote concerns functions defined on the closed upper half space (the subset of $\mathbb R^n$ with $x_n \ge 0$).
My question: Where can I find a proof of the quoted theorem?