Is $$ \left\{\left(x, \frac{1}{x}\cdot \cos\left( \frac{1}{x^2}\right)\right) \mid x \in \mathbb{R} \setminus \{0\}\right\} \cup \{ (0,0)\} \subset \mathbb{R^2}$$ a connected set?
I tried proving by contradiction that it is connected , but it didn't seem to lead anywhere. It would be easy to prove that it is not connected if it did not contain $(0,0)$.
$\{(x,\frac 1 x \cos(\frac 1 {x^{2}})): x>0\}$ is connected because it is a continuous image of a connected set. Now $(0,0)$ belongs to the closure of this set because $(0,0)=\lim (x_n,\frac 1 {x_n} \cos(\frac 1 {x_n^{2}}))$ where $x_n=((n+\frac 1 2)\pi)^{-1/2}$. Hence $\{(x,\frac 1 x \cos(\frac 1 {x^{2}})): x \geq0\}$ is connected and $\{(x,\frac 1 x \cos(\frac 1 {x^{2}})): x\leq 0\}$ is connected is connected by a similar argument. The union of these two is also connected because they have a point in common.