I need to find a continuous function $$ \ f:[0,1]\to D^2 $$
I know we can map a line to a circle using the inverse of stereographic projection. So, if I can find a map which can map each point in the closed interval to a line I can find such function using composition.
Can we continuously stretch a point to a line?
Is such a map even possible or am I just wasting my time?
Sure it's possible: the Cantor function is a surjective continuous function from $C$ (the Cantor set in $[0,1]$, or any space homeomorphic to it, like $\{0,1\}^{\Bbb N}$) onto $[0,1]$. Call it $c: C \to [0,1]$. Then $c_2 : C^2 \to [0,1]^2$, defined by $c_2(x,y)=(c(x), c(y))$ is also continuous and sujective.
Finally we use that $[0,1]^2 \simeq D^2$ and $C^2 \simeq C$ so we have a continuous onto map from $C$ onto $D^2$, that we can extend (by Tietze, or linear interpolation etc.) to a continuous map from $[0,1]$ onto $D^2$. It's not a nice simple formula but such maps do exist.
Fact: every Peano continuum (a metric compact, connected and locally connected space) is a continuous image of $[0,1]$. This is a classic theorem (not trivial at all, but useful to know).