Suppose we have a function $f(q) = q^2$ defined over the quaternions $\mathbb{H}$.
How does one formally show that the function $f$ defines on $\mathbb{R}^4 \cup \{\infty\} \cong S^4$ is smooth without specifying a smooth manifold structure on $\mathbb{R}^4 \cup \{\infty\}$ first?
On $\mathbb{R}^4$, $f$ clearly is a polynomial and therefore smooth. We only need a smooth extension to a "pole" on $S^4$. To show continuity, one could argue that as $\vec{x} \to \infty$, $\lvert f(\vec{x}) \rvert = \lvert \vec{x} \rvert^2 \to \infty$, and thus $f(\vec{x}) \to \infty$, but thus defining $f(\infty) = \infty$ would make the extension continuous. This is rather informal however. To show smoothness, I am stuck.
Thanks for any hints or solutions!