Path-Connected Sets and Their Closures: A Counterexample?

[Bashyboy]

Homework Statement

I am trying to determine whether the closure of a path-connected set is path-connected.

Homework Equations

The Attempt at a Solution

Let $S = \{(x, \sin(1/x) ~|~ x \in (0,1] \}$. Then the the closure of $S$ is the Topologist's Sine Curve, which is known not to be path-connected. However, recalling that the image of a path-connected space under a continuous function is path-connected, and defining $g : (0,1] \rightarrow \mathbb{R}$ as $g(x) = (x, \sin (1/x))$, we see that $S = g((0,1])$ must be a path-connected space.

My question is, would this constitute a counterexample to the claim, or have I made some error?

 

[zwierz]

there is no error I guess