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

A Jordan curve is a continuous closed curve in $\Bbb R^2$ which is simple, i.e. has no 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. My question is, 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.

EDIT: As @dfeuer suggested, let's also require that the curve goes off to infinity in both directions. To make this precise, let's say that there exists two lines $L_1$ and $L_2$, parametrized by $L_1(t) = (a_1 + b_1 t, c_1 + d_1 t)$ and $L_2(t) = (a_2 + b_2 t, c_2 + d_2 t)$, such that the limit of $d(f(t), L_1(t))$ as $t$ goes to $-\infty$ is $0$, and the limit of $d(f(t), L_2(t))$ as $t$ goes to $\infty$ is $0$. Under that condition, does the complement of the curve have two connected components?

Solution 1:

Consider the curve $\gamma:t\mapsto(e^t,e^{-t}\sin(e^{-t}))$, this maps $\Bbb R$ homeomorphically to the graph $\Gamma$ of the map $x\mapsto \frac1x\cdot\sin\left(\frac1x\right)$. The image of $(-\infty,0)$ is oscillating with increasing amplitude towards $0$, and the image of $(0,\infty)$ is clearly unbounded in $x$-direction.

Define $$C_+=\left\{(x,y)\mid x>0,y>\frac1x\sin\left(\frac1x\right)\right\}\\C_-=\left\{(x,y)\mid x>0,y<\frac1x\sin\left(\frac1x\right)\right\}$$ and $$C_0=(-\infty,0]\times\Bbb R$$

It is easy to prove that all $C'$s, whose union is the complement of $\Gamma$, are path-connected and no point in one of these sets can be joined to a point in another via a path not intersecting $\Gamma$.

On the other hand, since connected components are closed (in $\Bbb R^2-\Gamma$) and $(0,y)\in C_0$ is in the boundary of $C_+$ and $C_-$ (for arbitrary $y$), it follows that $C_-$ and $C_+$ are not the connected components of $\Bbb R^2-\Gamma$. Hence there is only one component in the complement.

Components of $\Bbb R^2-\text{Im}(\gamma)$: $\quad 1$

Path-components: $\qquad\qquad\,\quad 3$