Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$? 
Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$ ?

I tried writing it as a union of two connected sets , or otherwise as a union of two disjoint non-empty open sets without any success , Please help . Thanks in advance 

The image of $A$ under the continuous projection $(x, y) \to x$ is $\mathbb{R}\setminus\{0\}$, which is not connected. Hence $A$ itself is not connected.
