Interiors of continuous images of the middle-third Cantor set

Abstract

Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the image \begin{equation*} f_{U}(C,C)=\{f(x,y):(x,y)\in (C\times C)\cap U\}. \end{equation*} If $\partial _{x}f$, $\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\in (C\times C)\cap U$ such that \begin{equation*} 1<\left\vert \frac{\partial _{x}f|_{(x_{0},y_{0})}}{\partial _{y}f|_{(x_{0},y_{0})}}\right\vert <3\text{ or }1<\left\vert \frac{\partial _{y}f|_{(x_{0},y_{0})}}{\partial _{x}f|_{(x_{0},y_{0})}}\right\vert <3, \end{equation*} then $f_{U}(C,C)$ has a non-empty interior. As a consequence, if \begin{equation*} f(x,y)=x^{\alpha }y^{\beta }(\alpha \beta \neq 0),\text{ }x^{\alpha }\pm y^{\alpha }(\alpha \neq 0)\text{ or }\sin (x)\cos (y), \end{equation*} then $f_{U}(C,C)$ contains a non-empty interior.