Boundary of a compact connected set in $\Bbb R^2$ If I have a compact connected set in $\Bbb R^2$, and I'm examining the boundary points of this set. Is it true that around every cusp/corner on the boundary, there's an open interval where the boundary is smooth? This seems intuitive to me, but I don't know if it's true.
I.e.: is it possible that the boundary is somehow badly behaved everywhere?

Take a function $f : [0,1]\to \mathbb R$ which is continuous but nowhere differentiable with $f\geq 0, f(0)=f(1)=0$, and take $\{(x,y) \mid x\in [0,1] \land -f(x)\leq y \leq f(x)\}$.
This is clearly connected and compact. But the boundary is not smooth at any point, as the boundary is precisely the union of the graphs of $f$ and $-f$
