${\mathbb{R}}^{2} $ **Euclidean 2-space**,Let $\mathit {X} \subset \mathbb{R}^{2} $.$$\mathit {X}=[-2,2]\times[-1,0]\cup[-2,-1]\times[0,1]\cup[1,2]\times[0,1]$$
$\qquad\qquad\qquad\qquad\qquad$ ![enter image description here](https://i.stack.imgur.com/N600R.png)

From the geometric intuition, we know $ \mathit {X}$ is ***path connected***. I want to prove it theoretically.So I tied to use **Definition**: ***A topological space $\mathit {X} $ is path connected if for every $x,y\in \mathit {X}$ there is a path in $\mathit {X}$ from $x$ to $y$***.but for every $x,y\in \mathit{X}$, finding a continuous function $f:[0,1]\rightarrow \mathit{X}$ such that $f(0)=x$ and $f(1)=y$ is seemingly difficulty.
I need some help to go further or some other solutions to prove it. Any of your help will be appreciated!