I took a countable dense set in R^n - C, x_j, j = 1, 2, ..., and for each j took the closed circle with center x_j and radius d(x_j, C) (distance from x_j to C

I took a countable dense set in R^n - C, x_j, j = 1, 2, ..., and for each j took the closed circle with center x_j and radius d(x_j, C) (distance from x_j to C which is well defined due to compactness). Then I defined a function g_j: R^n --> R 0 outside the circle and on the circle with values on the radii much as in the function Milnor, Rudin, and I mentioned. So, g_j is infinitely differentiable, strictly positive on the interior of the circle at x_j, and 0 otherwise. Then I added the functions g_j in a convergent way, again using the exponential trick. The resulting function was as desired, 0 on C, strictly positive otherwise, and infinitely differentiable.