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 

Recall that a connected and locally path-connected topological space is path-connected. Your subspace $X=\left\{\left(x,\sin\frac1x\right)\,:\,x\ne0\right\}$ is locally path-connected. However, you can easily convince yourself (in fact, if you asked the question, you probably already are) that no curve with image contained in $X$ joins $\left(\frac1\pi,0\right)$ with $\left(-\frac1\pi,0\right)$.
