Smoothness isn't necessary to prove the result, just continuity. You do need smoothness to prove a bound on the rate of convergence of the basis representation

Smoothness isn't necessary to prove the result, just continuity. You do need smoothness to prove a bound on the rate of convergence of the basis representation, and given that the boundary of the Mandelbrot doesn't have a closed form representation (as far as I'm aware), I think that the convergence of the neural network representation would be extremely slow.