Let $S=\{(x,y,z)\in \Bbb R^3: x^2+y^2=z^2 \wedge 0\leq z\leq 4\}$.
I'd like to create a function $\vec \Sigma:A\subseteq \Bbb R\to \Bbb R^3/\vec \Sigma(t)=(x(t),y(t),z(t))$ and $\text{Im}(\vec\Sigma)=S$.
I know such a function exists because $|\Bbb R|=|S|$, and I also believe that this function must be discontinuous, but either way I'd like to find it.
How could I do this?
You can do this continuously if you don't insist your mapping be a bijection: Let $c:[0, 1] \to [0, 1] \times [0, 1]$ denote your favorite continuous surjection, write $c(t) = \bigl(u(t), v(t)\bigr)$, and define $$ \Sigma(t) = \bigl(4u(t) \cos 2\pi v(t), 4u(t) \sin 2\pi v(t), 4u(t)\bigr). $$