Show that the region $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is not path connected.
Suppose that $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is path connected. Then we can define a path $L$ from $(a,y)$ to $(b,y)$, where $a < 0$, $b >1$ and $y$ is arbitrary in $\mathbb{R}^2$ such that $$L = \{(a,b) : (1-t)a+tb, t \in [0,1] \}.$$
My issue is that I've taken a particular path, the result is obvious, but I'm unsure of how to prove it rigorously.
The projection $\pi$ on the $x$-axis is a continuous map, hence assuming that there is a path $\gamma$ between $(-2,0)$ and $(2,0)$, there is an interval (given by $\pi(\gamma)$) enclosing $-2$ and $2$ in $\pi(X)$. However, $-2$ and $2$ belong to two different connected components of $\pi(X)$.