The typical structure of the sets $\{x\colon \;f(x)=h(x)\}$ for $f$ continuous and $h$ Lipschitz

Abstract:

Let $R$ be the space of real numbers and $C$ the space of continuous functions $f:[0,1] \to R$ with the uniform norm. Bruckner and Garg prove that there exists a residual set $B$ in $C$ such that for every function $f \in B$ there exists a countable dense set ${\Lambda _f}$ in $R$ such that: for $\lambda \notin {\Lambda _f}$ the top and bottom levels in the direction $\lambda$ of $f$ are singletons, in between these levels there are countably many levels in the direction $\lambda$ of $f$ that consist of a nonempty perfect set together with a single isolated point, and the remaining levels in the direction $\lambda$ of $f$ are all perfect; for $\lambda \in {\Lambda _f}$ the level structure in the direction $\lambda$ of $f$ is the same except that one (and only one) of the levels has two isolated points instead of one. In this paper we show that the analogue of the above theorem holds: if we replace the family of straight lines $\{ \lambda x + c\}$ by a $2$-parameter family $H$ that is almost uniformly Lipschitz; and if we replace $\{ \lambda x + c\}$ by a homeomorphical image of a certain $2$-parameter family $H$ that is almost uniformly Lipschitz.