Rudin Ex 8.6

Rudin Ex 8.7 (Rudin's $D_i f = \partial f / \partial x_i$)

Show that, for any closed subset $E \In \R^2$, there is a continuous function $f: \R^2 \to \R$, such that $f^{-1}(0) = E$. (bonus, can you make $f$ a smooth function?)

For the implicit function theorem, take $n=m=1$, and interpret it graphically and intuitively.