Is the following set path-connected?

Question

Is the following set path-connected?

$A=\{(x,y):y=x\sin \frac{1}{x},x>0\}$

I am unable to understand that should I prove it or disprove it .

Will someone please give me some hints.

Answer 1

This is the graph of a function which is continuous on the domain specified, and so the graph is homeomorphic to its domain. Therefore it is path connected. For any two points, just map the interval between the $x$ coordinates into the plane with the function.

It is closely related to the topologist's sine curve, which is connected but not path connected. To obtain that space you have to add a vertical interval at $x=0$.

Answer 2

Hint: $y$ is defined everywhere for $x>0$, there are no special points, as x cannot be zero.