Question

This isn't a proof, but I always liked Brownian motion as an example of a continuous, nowhere differentiable function. As opposed to putting a lot of effort into artificially creating a series which converges and proving that it is continuous and nowhere differentiable, you take a natural, physical process and say "Ah ha! It works." The proof that it's nowhere differentiable is a pretty straightforward, though constructing a Brownian motion can be a pain...