Beauty and the Beast: Smooth meets the Monster. Let n be a positive integer and let R be the set of real numbers. Let R^n be Euclidean n-dimensional space.

Beauty and the Beast: Smooth meets the Monster. Let n be a positive integer and let R be the set of real numbers. Let R^n be Euclidean n-dimensional space. E.g., R^1 is the real line; R^2 is the plane; and R^3 is (approximately) the space we live in. Let C be a closed subset of R^n. In R^2, examples of closed subsets include a sample path of Brownian motion (as in the OP) and the Mandelbrot set. An example in R^1 is a Cantor set of positive measure. Then there exists a function f: R^n --> R so that (1) f is infinitely differentiable, (2) f(x) = 0 for all x in C, and (3) f(x) > 0 for all x not in C. So, smooth beauty f meets the monster beast set C. Of course, the level set of any differentiable function is closed. But now we know that any closed set can be the level set of an infinitely differentiable function. So, we also know that there can be an infinitely differentiable function positive outside of C, 0 on the boundary of C, and negative on the interior of C. So, we know that for any closed set, there is an infinitely differentiable function that has the boundary of the closed set as a level set. Since a level set of a differentiable function is closed, we also have the the boundary of any closed set is closed. "I have found a truly wonderful proof," but the mathematical notation is too difficult to type into a blog post! But, no worries: I published it in JOTA. The Brownian motion example was noticed by A. Karr.