The extension of smooth function If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an arbitrary domain $V \supseteq \bar U$, is there a smooth function $\tilde f$ on $V$ extending $f$?
 A: We can take $V=\mathbb R^n$ without losing anything. The answer is yes, but this is nontrivial and I'm not going to prove it here. Here are some sources:
1) Short paper by Seeley (1964) covers the case of half-space. If you are interested in local matters, then straighten out a piece of $\partial U$ and apply this reflection-based extension.
http://www.ams.org/journals/proc/1964-015-04/S0002-9939-1964-0165392-8/home.html
2) Whitney's classic paper of 1934 treats general closed sets (!) but is not an easy reading.
http://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/home.html
3) In between there was a paper by M.R. Hestenes, "Extension of the range of a differentiable function" (Duke Math. J. 8, (1941) 183--192). Unfortunately, Duke is less generous with old articles than the AMS; the article is behinds a paywall.
4) A very recent 2-volume book by A. Brudnyi and Yu. Brudnyi "Methods of Geometric Analysis in Extension and Trace Problems" is an encyclopedia of extension theorems. See Chapter 2 of volume 1.
