Interiors of continuous images of the middle-third Cantor set

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 $\partial {x}f$,$\partial {y}f$arecontinuouson$U,$andthereisapoint$(x{0},y{0})\in (C\times C)\cap U$ 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.