I am much more familiar with math.SE. On that site there are explicit guidelines about LaTeX in titles and things like \displaystyle, \dfrac, \dbinom and similar are explicitly discouraged. See Guidelines for good use of $\LaTeX$ in question titles and perhaps also Why no use displaystyle in titl...

Write $g$ as the inverse of $f$. Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that $\displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0$ as $r\to0$? If you like, add more conditions - including that for each $x\in C$ there exists $r_x>0$ such that for any $\epsilon>...