Which of the following subsets are of classe $C^1$?

Question

I have the following subsets and I have to find which of these subsets are of class $C^1$ and, for those they are, find the exterior normal.

$\Omega_1 := \{(x, y) \in \mathbb{R}^2 : 1<x^2 + y^2 <9 \, \, \textbf{and} \, \, (x-2)^2 + (y-2)^2 > \frac{1}{4} \}$

$\Omega_2 := \{z \in \mathbb{R}^2 : \exists \, k \geq 1 \, \, \text{such that} \, \, | z-z_k| < 2^{-3k} \}$, where $z_k = (2^{-k}, 0) \in \mathbb{R}^2$

$\Omega_3 := \{(r \cos(\theta), r \sin(\theta)) \in \mathbb{R}^2 : \theta \in \mathbb{R}, 0 \leq r < f(\theta) \}$, where $\, f: \mathbb{R} \rightarrow (0, \infty) $ is a continuously differentiable function with $f(r+ 2\pi)=f(r) \, \forall \, \, r \in \mathbb{R}$.

Intuitively the first set $\Omega_1$is not of class $C^1$ because it has two bad intersection points, but I have no idea to prove it rigorously. And for the other two sets I really have some difficulties to see if they'are or not.

Any suggestions? Thanks in advance!

Answer 1

Not the same set, but same idea: let be $C = \{(x,0):x\ge 0\}\cup\{(0,y):y\ge 0 \}$. Suppose there is a $C^1$ function $$\gamma = (\gamma_1,\gamma_2):(-1,1)\longrightarrow\Bbb R^2$$ with

• $\gamma(0) = (0,0)$,
• $\forall t\in(-1,1): \gamma'(t) = 1$ (the differential has rank 1 in every point),
• $\gamma(-1,1) = C\,\cap\,$(some open set of $\Bbb R^2$).

Consider the first coordinate of $\gamma $. By the third condition: $$\forall t\in(-1,1): \gamma_1(t)\ge 0 = \gamma_1(0)$$. I.e., $0$ is a local minimum of $\gamma_1$, so $\gamma_1'(0) = 0$. By the same reason, $\gamma_2'(0) = 0$ and $\gamma(0) = (0,0)$ in contradiction with the second condition.