Example: For positive integer n and the set R of real numbers, for any closed set C in R^n, there exists a function f: R^n --> R so that f(x) = 0 for all x in C, f(x) > 0 otherwise, and f is infinitely differentiable.

I have a hunch that this can be solved similarly to the construction of bump function on a given compact set [4], except the support is an open set and I'm not sure what the "appropriate scaling" should look like.

[4] https://en.wikipedia.org/wiki/Bump_function#Existence_of_bum...