Does the Jordan curve theorem apply to non-closed curves?

Does the Jordan curve theorem apply to non-closed curves?

A Jordan curve is a continuous closed curve in $\Bbb R^2$ without
self-intersections. The Jordan curve theorem states that the complement of
any Jordan curve has two connected components, an interior and an
exterior.
Now let's define an unbounded curve to be a continuous map $f:
(-\infty,\infty)\to\Bbb R^2$ such that $f((-\infty,0))$ and
$f((0,\infty))$ are both unbounded. Then does the complement of a simple
unbounded curve always have two connected components? It seems intuitively
true, since you'd expect the curve to have two sides, but considering how
long it took to prove the Jordan curve theorem, things may not be as
straightforward as they appear.
Any help would be greatly appreciated.
Thank You in Advance