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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?

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up vote 6 down vote accepted

Good question. One direction is immediate; the other direction depends on a lemma of Émile Borel, which shows that there is a smooth function defined in the lower half-space whose derivatives all match those of $F$ on $U \smallsetminus \mathbb H^n$. See, e.g., Theorem 1.2.6 in The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., by Lars Hörmander. (I included this reference in the second edition of my Smooth Manifolds book.)

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