Molehills out of mountains

Tom W. Korner, Cambridge University. Part of the analysis and applications seminar series.

We construct a non-constant, once differentiable function

$f:[0,1]^{2}\rightarrow{\mathbb R}$ such that given any two points ${\mathbf a},\,{\mathbf b}\in[0,1]^{2}$  we can find a continuous $\gamma:[0,1]\rightarrow [0,1]^{2}$ such that $\gamma(0)={\mathbf a}$, $\gamma(1)={\mathbf b}$ and $f$ has zero derivative at ever $\gamma(t)$ with $t\neq 0,\,1$.

The tools used come from Calculus II (or, possibly, III)

Tom W. Korner, Cambridge University