Are there any continuous functions for the Zariski topology other than polynomials?

Question

Let $X$ be an affine algebraic subset of the $n$-dimensional affine space $A^n$ over a field $k$, and $A$ the affine line, both equipped with the Zariski topology. Is it true that a continuous function $f: X\rightarrow A$ is always the restriction of a polynomial in $k[x_1,...,x_n]$ to $X$?

Answer 1

The topology on the affine line is the cofinite topology, so any bijection $\mathbb{A}^1\to\mathbb{A}^1$ is continuous. Certainly not all of these are given by polynomials, in general.