414/514 – The Schoenberg functions

Here is Jeremy Ryder’s project from last term, on the Schoenberg functions. Here we have a space-filling continuous map $f:x\mapsto(\phi_s(x),\psi_s(x))$ whose coordinate functions $\phi_s$ and $\psi_s$ are nowhere differentiable.

The proof that $\phi_s,\psi_s$ are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass $M$-test applies.

The proof that $f$ 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 $f$ can be seen in page 49 of Thim’s master thesis.