Why $\{(x,y)\in\Bbb{R}^2 :\, y=x\sin \frac1x\}\cup\{(0,0)\}$ is connected but not compact?

Question

I wonder why the set of all points in the plane satisfying $y = x\sin \frac {1}{ x}$ together with the origin is connected but not compact.

Is there any example of a open cover that is not finite?

Answer 1

This set is connected because this is the graph of a continuous function. Namely, $g(x)=x\sin\frac{1}{x}$ for $x\ne 0$ and $g(0)=0$ is a continuous function.

Answer 2

That set is an unbounded subset of $\mathbb R^2$, and therefore it is not compact.