Construct an explicit bijective map $ f: D \to \overline{D} $, where
$$D := \left\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\right\}$$
is the open unit disk in $ \mathbb{R}^2 $, and $ \overline{D}$ is the closure of $ D $, that is,
$$\overline{D} := \left\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\right\}.$$
I considered the similar problem that the bijection from $(0,1]$ to $[0,1]$.$$(0,1]\longrightarrow[0,1]$$$$1\longrightarrow0$$$$\frac{1}{2}\longrightarrow1$$$$\frac{1}{3}\longrightarrow\frac{1}{2}$$$$\cdots$$However, in this case, I cannot map every point on $\partial D$ to origin so that it is not a bijection.