Neat.  Did you do it by making such a function for the open ball, and then, given an open set defined as a countable union of open balls, define a sequence of such functions scaled appropriately so that their derivatives all converge uniformly?<p>Edit: Looks like you can make that work, according to the answer at <a href="https:&#x2F;&#x2F;math.stackexchange.com&#x2F;questions&#x2F;791248&#x2F;every-closed-subset-e-subseteq-mathbbrn-is-the-zero-point-set-of-a-smooth" rel="nofollow">https:&#x2F;&#x2F;math.stackexchange.com&#x2F;questions&#x2F;791248&#x2F;every-closed...</a>