Interiors of continuous images of the middle-third Cantor set

arXiv1809.01880MaRDI QIDQ6306322

Kan Jiang, Lifeng Xi

Publication date: 6 September 2018

Abstract: Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $Usubsetmathbb{R}^2$. Denote the image �egin{equation*} f_{U}(C,C)={f(x,y):(x,y)in (C imes C)cap U}. end{equation*} If $partia{l}_{x}f$, $partia{l}_{y}f$ are continuous on $U,$ and there is a point $(x_0,y_0)in(CimesC)capU$ such that �egin{equation*} 1<leftvert frac{partial _{x}f|_{(x_{0},y_{0})}}{partial _{y}f|_{(x_{0},y_{0})}} ightvert <3 ext{ or }1<leftvert frac{partial _{y}f|_{(x_{0},y_{0})}}{partial _{x}f|_{(x_{0},y_{0})}} ightvert <3, end{equation*} then $f_U(C,C)$ has a non-empty interior. As a consequence, if �egin{equation*} f(x,y)=x^{alpha }y^{�eta }(alpha �eta

eq 0), ext{ }x^{alpha }pm y^{alpha }(alpha

eq 0) ext{ or }sin (x)cos (y), end{equation*} then $f_U(C,C)$ contains a non-empty interior.

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