Line segments intersecting Jordan curve I have thought this problem a week without success.
Is there a set $A\subset \mathbb{R}^2$ such that 


*

*The boundary of $A$, $\partial A$, is a Jordan curve and

*For any $B\in \operatorname{int} A\ne\emptyset $, $C\in \operatorname{ext} A\ne\emptyset$ , the line segment $BC$ intersects $\partial A$ infinitely many times?


Any ideas?

I am not sure, but here's an idea. Take the graph of $x sin(1/x)$ and join the two zeroes around (+/-)0.15 by a big enough curve. Take the points to be $(0.12, 0)$ and $(-100, 0)$.
EDIT: Misunderstood, that it is required for only a single pair of points. OP wants this to happen for every pair of points.
