Are the subsets $G^{+}$ and $G^{-}$ connected in $\mathbb{R^{2}}\setminus G$?
It's instructive to write $G$ as the image of a continuous function. Namely,
$$G = g_0(\mathbb{R});\,\, g_{0}(x) = (x, f(x))$$
Consider extending to $g_{c}(x) = (x, f(x)+c)$. Then $g$ is a continuous bijection $\mathbb{R}^2 \to \mathbb{R}^2$. Clearly $G^{+}= g(\mathbb{R} \times (0, \infty))$ and ditto for $G^{-}$. They are connected sets, so their images under $g$ are connected too.