# Finding Limit Points, Interior Points, Isolated Points, the Closure of

Finding Limit Points, Interior Points, Isolated Points, the Closure of $A \subset \mathbb{R}^2$, where $A$ is the graph of the function $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x)= \sin(1/x)$ if $x$ doesn't equal $0$ and $0$ if $x=0$. (The distance in $\mathbb{R}^2$ is the standard $d_2$.

I am completely unsure how to approach the problem.

I believe that there is no limit points because when I take the limit of $\sin(1/x)$ as $x \to \infty$ I get that the function is jumping between 1 and -1. Is this right?

• not sure what you meant exactly, particularly with \sub, but please check and correct... Commented Feb 10, 2014 at 21:46

I suggest you graph the function, and check what happens as $x \rightarrow \pm \infty$ (can you explain this behavior?). To find the closure of $A$, find all the points that the graph approaches arbitrarily closely. These are your boundary points, and the closure is the interior of $A$ along with the boundary of $A$.