How do I proof that the interior of C is empty?

Question

I have had trouble proving that the interior of the following set is empty. I have tried to do it by definition, but haven't managed to figure out the proof.

$$ C := \{ (x,y) \in \mathbb{R^2} \mid x \in (-1,1) \text{ and } y=x^3 \} $$

Answer 1

If the ball of radius $r$ around point $(a,b)$ is $\subset C$, then $(a,b\pm\frac r2)$ must be $\in C$

Answer 2

As a hint try to show that for any $(x_0,y_0)\in C$ there doesn't exist any $\epsilon>0$ such that$$B_\epsilon(x_0,y_0)\in C$$where$$B_\epsilon(x_0,y_0)=\{(x,y)\in\Bbb R^2|\sqrt{(x-x_0)^2+(y-y_0)^2}<\epsilon\}$$

Answer 3

Suppose that the interior is non-empty, then there is a ball $B$ with radius $r>0$ around some point $(p,q)$ of $C$ such that $B \subseteq C$.

So we know that $q= p^3$ and $p \in (-1,1)$ from $(p,q) \in C$.

Now $(p,q+\frac{r}{2}) \in B$ because $$\| (p,q) - (p, q+\frac{r}{2})\| = \| (0,\frac{r}{2})\| = \frac{r}{2} < r$$ where $\|\cdot\|$ denotes the distance/norm in the plane.

So by our assumption $B \subseteq C$ we would have $(p, q+\frac{r}{2}) \in C$.

But $(q+\frac{r}{2})^3 = q^3 + 3\frac{r}{2}q^2 + 3\frac{r^2}{4}q + \frac{r^3}{8} \neq q^3 = p$ so that the point is not in $C$, contradiction.

(it's intuitively clear: $C$ is part of a curve and any open ball based on a point of it sticks out because we have other $y$ coordinates lying on a small distance...)

So the interior of $C$ is empty.

Answer 4

To show the interior of $C$ is empty, it is enough to show that there is not point of $\mathbb R^2$ that is both in $C$ and an interior point of $C$. So it is enough to show that any point of $C$ is not an interior point.

So if $(a,b) \in C$ it is enough to show that for any radius $r$ the open ball around $(a,b)$, the $B_r(a,b)$ will contain a point that is not in $C$.

Now $d((a,b), (a, b+\frac r2)) = \sqrt{ (a -a)^2 + (b+\frac r2 -b)^2} = \frac r2 <r$ so $(a,b+ \frac r2)\in B_r(a,b)$.

But $(a,b) \in C$ so $b = a^3$ so $b + \frac r2 \ne a^3$. So $(a,b+\frac r2) \not \in C$.

So $(a,b+\frac r2)$ is a point that is in $B_r(a,b)$ but is not in $C$ and as $r$ could be any distance, $(a,b)$ is not an interior point, and as $(a,b)$ could be any point of $C$ no point of $C$ is an interior point, so there are not points in $C$ that are interior points, so $C$ has empty interior.