I have seen a number of proofs on this site about the existence or lack of a retraction from $\Bbb R^2$ to $S^1$. A common point for these discussions is about the relationship between the unit circle $S^1$ and the closed disk $D$ as parts of the entire plane $\Bbb R^2$. I can understand the Brouwer approach for disproving such a retraction. However, I can't figure out why a retraction from $\Bbb R^2$ to $D$ is being taken for granted. (I.e. I couldn't prove this off hand and so would feel hesitant to use it in a proof)
Am I missing a proof for the $\Bbb R^2$ to $D$ retraction somewhere online? I looked pretty extensively.
If not, does anyone have suggestions for showing there is a retraction from $\Bbb R^2$ to the closed disk $D$?
For clarity, I'm defining $D =\lbrace (x,y) \in \Bbb R^2 \mid x^2 + y^2 \leq 1 \rbrace $
Such a retraction is taken for granted since one can simply define$$\begin{array}{ccc}\Bbb R^2&\longrightarrow&D\\v&\mapsto&\dfrac v{\max\{1,\|v\|\}}.\end{array}$$Note that this function is continuous, since the norm is continuous and the function$$\begin{array}{ccc}\Bbb R&\longrightarrow&\Bbb R\\x&\mapsto&\max\{1,x\}\end{array}$$is continuous too.