Let $\sim$ denote the equivalence relation on the $n$-sphere $S^n$ defined via $$ (x_1,\dots,x_n,x_{n+1})\sim(x_1,\dots,x_n,−x_{n+1})\:\text{ for all }\: (x_1,\dots,x_{n+1})\in S^n. $$
Show that the quotient space $S^n/\sim$ is homeomorphic to the $n$-disc $D^n$.
I know that Ii have to show that there exists a bijective function $f: (S^n/\sim) \to D^n$ that is continuous and that the pre-image $f^{-1}$ is continuous as well, but I can't advance from here, any tips?
Hint: Define$$\begin{array}{rccc}f\colon&S^n/\sim&\longrightarrow&D^n\\&\bigl[(x_1,\ldots,x_n,x_{n+1})\bigr]&\mapsto&(x_1,\ldots,x_n).\end{array}$$