Reffering to this question Clarke's tangent cone, Bouligand's tangent cone, and set regularity I'm asking myself if may exist a closed bounded set $S\in\mathbb{R}^2$ and a point $x\in\partial S$ such that the two cones (Bouligand and Clarke cones) reduces both to $\{0\}$, namely only the origin. Is this possible if the boundary is piecewise the graph of a continuous function?
I think the answer has to be no, at least for the second part of the question but I'm not able to find a proof.
Any reference to books or articles will be greatly appreciated.
Thank you in advance.
It might depend on your notion of "piecewise the graph of a continuous function".
Define $r \colon [0,1] \to [0,1/e]$ via $r(\theta) = \exp(-1/\theta)$ for $\theta > 0$ and $r(0) = 0$. Then $r$ is continuous and $$ C := \{ r(\theta) (\cos(\theta),\sin(\theta)) \mid \theta \in [0,1]\} \subset \mathbb R^2 $$ is the image of a continuous function. Then, a straightforward but tedious computation yields that both cones at $0$ reduce to $\{0\}$.