Here is Jeremy Ryder’s project from last term, on the Schoenberg functions. Here we have a space-filling continuous map whose coordinate functions and are nowhere differentiable.
The proof that are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass -test applies.
The proof that is space filling is nice and short. The original argument can be downloaded here. A nice graph of the first few stages of the infinite fractal-like process that leads to the graph of can be seen in page 49 of Thim’s master thesis.