Interior of the Graph of $\sin(1/x)$ Let $f(x)=\sin(1/x)$. If I'm correct then the boundary of $S=\{(x,f(x)):x\in\mathbb{R}\} \subset \mathbb{R}^2$ is $\{(x,f(x)):x \text{ in domain of } f(x)\}\cup\{(0,0)\}$, since for any $\varepsilon > 0$ and an open ball $B_\varepsilon(0)$, $B_\varepsilon(0)\cap S\ne \emptyset \ne B_\varepsilon(0)\cap [S]^c$. [I have some doubts about this part though].
For the interior of $S$, it is the empty set, since there does not exist an $\varepsilon > 0$ such that $B_\varepsilon((x,f(x)))\subset S$.
Thus, the closure of $S$ is $S\cup\{(0,0)\}$.
Please let me know if you think this is about right.
 A: As a bit of notational touching-up, the graph of $f$ is the set $G:=\{(x,f(x)) \; : \; x \text{ in the domain of } f\} \subset \mathbb{R}^2$.
You're correct that the interior of $G$ is empty, and the boundary of $G$ is equal to the closure of $G$.
However, I claim your closure is incorrect. It should be $G\cup\left(\{0\}\times[-1,1]\right)$ (that is, add in the whole vertical line segment from (0,-1) to (0,1) ). Your argument justifying that this is the closure/boundary should work for this larger set too, although you should change the notation for your open balls' centers to reflect the fact that you have a coordinate pair: $B_{\varepsilon}(0,y)$.
A: This looks generally right. The interior of the graph is empty, though it's boundary is $S\cup\{(0,y):y\in[-1,1]\}$. This might have been what you meant the whole time and I made a mistake when correcting your notation.
To see that this whole segment is contained in the boundary, consider a horizontal line in the plane that intersects the graph. The intersection of this line with the graph creates a sequence of points that converge to the intersection of the line with the y-axis. Alternatively, your argument holds for this extended set. Either way, you still need to argue that no other point is in the boundary. This can be done by showing that the set is closed, as closure is monotonic and a closed set is its own closure.
A: see this one: Topologist's sine curve https://en.wikipedia.org/wiki/Topologist%27s_sine_curve
