I want to find some characterization of the continuous functions from $I=[0,1]$ to $\mathbb{A}^n$ where $\mathbb{A}^n,n\in\mathbb{N}$ is an affine space (with the Zariski topology) over some field $F$ (If it's necessary, $\mathbb{R}$ or $\mathbb{C}$).
I thought about using this approach:
If $f$ is such a function, the inverse image of any algebraic set is a closed subset of $[0,1]$. But I don't seem to arrive anywhere.
By the way, I don't need to characterize all continuous functions, only the ones such that $f(0)=f(1)$.
So... does this approach work, or would there be a better one? Or is it too hard to characterize these functions?
The motivation for this is finding the fundamental group of some affine space.
Since the Zariski topology on $\mathbb A^n$ for the field of real or complex numbers is much more coarse than the Euclidean topology, there will be a lot more continuousloops (continuous maps from $[0,1] $ to $\mathbb A^n$ with $f(0)=f(1)$).
However, the same argument that shows that $\pi_1(\mathbb R^n,x_0)$ and $\pi_1(\mathbb C^n,x_0)$ are trivial (with the Euclidean topology) shows this is also the case for $\pi_1(\mathbb A^n,x_0)$. Namely, given any loop $\gamma$ based at $x_0$, $(1-t)\gamma$ provides a homotopy between it and the trivial loop based at $x_0$. It does not take much work to see that for each $t\in[0,1]$, $(1-t)\gamma$ is a continuous loop.
Note that this argument only works for $\mathbb R,\mathbb C$ because we are using completeness quite strongly.