Construction of a vector valued function that maps from $\mathbb{R}^n$ to an embedded disc

Question

Does there exist a continuous, surjective, vector valued function

$f: \mathbb{R}^n \to B$

where $B$ is the disc embedded in the positive portion of $\mathbb{R}^n$

$B = \{x \in \mathbb{R}^n| x_i \geq 0, x_1^2 + \ldots + x_n^2 = 1\}$

Answer 1

By the gluing lemma, the map

$$ g : x \in \mathbb{R}^n \mapsto \begin{cases}x &\text{if $\|x\| \leq 1$}\\\|x\|^{-1}x&\text{if $\|x\| \geq 1$}\end{cases} \in \mathbb{R}^n $$

is continuous and its image lies in the disc $\mathbb{D}^n = \{p \in \mathbb{R}^n: \|p\| \leq 1\}$. On the other hand, since $| \cdot | : t \in \mathbb{R} \mapsto |t| \in \mathbb{R}$ is continuous, we have that

$$ h = | \cdot | \times \cdots \times | \cdot | : (x_1,\dots,x_n) \in \mathbb{R}^n \mapsto (|x_1|, \dots,|x_n|) \in \mathbb{R}^n $$

is continuous. Now, the function $f = hg : \mathbb{R}^n \to \mathbb{R}^n$ is continuous and $f(\mathbb{R}^n) = B$, it suffices to corestrict it to $B$ to obtain the desired mapping.