A counterexample to access theorems for $C^ \infty$ functions

Abstract:

We construct a function $f:{\mathbb {R}^2} \to \mathbb {R}$ with the following properties: (1) f is of class $\infty$. (2) If $m \in {\mathbb {R}^2}$, then $\gamma :[0,1] \to {\mathbb {R}^2}$ exists such that $\gamma (0) = m,\gamma$ is continuous, and $f \circ \gamma$ is strictly increasing on [0, 1]. (3) If $\sigma :[0,1) \to {\mathbb {R}^2}$ is continuous and $f \circ \sigma$ is nondecreasing on [0, 1), then $\sup \{ |\sigma (s)|:0 \leq s < 1\} < \infty$.