Interiors of continuous images of self-similar sets with overlaps

Let $K$ be the attractor of the following iterated function system $$ \{S_{1}(x)=\lambda x,S_{2}(x)=\lambda x+c-\lambda,S_{3}(x)=\lambda x+1-\lambda\}, $$ where $S_{1}(I)\cap S_{2}(I)\neq\emptyset,(S_{1}(I)\cup S_{2}(I))\cap S_{3}(I)=\emptyset$, and $I=[0,1]$ is the convex hull of $K$. Let $d_{1}=\dfrac{1-c-\lambda}{\lambda}<\dfrac{1}{1-c-\lambda}=d_{2}$. Suppose that $f$ is a continuous function defined on an open set $U\subset\mathbb{R}^{2}$. Denote the image $$ f_{U}(K,K)=\{f(x,y):(x,y)\in(K\times K)\cap U\}. $$ If $\partial_{x}f$, $\partial_{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\in(K\times K)\cap U$ such that $$ \left\vert\frac{\partial_{y}f|_{(x_{0},y_{0})}}{\partial_{x}f|_{(x_{0},y_{0})}}\right\vert\in(d_{1},d_{2})\quad\text{ or }\quad\left\vert \frac{\partial_{x}f|_{(x_{0},y_{0})}}{\partial_{y}f|_{(x_{0},y_{0})}}\right\vert\in(d_{1},d_{2}), $$ then $f_{U}(K,K)$ contains an interval. As a result, we let $c=\lambda=\dfrac{1}{3}$, and if $$ f(x,y)=x^{\alpha}y^{\beta}(\alpha\beta\neq 0),\quad x^{\alpha}\pm y^{\alpha}(\alpha\neq 0),\quad\sin(x)\cos(y), \quad\text{ or } \, x\sin (xy), $$ then $f_{U}(C,C)$ contains an interval, where $C$ is the middle-third Cantor set.