I am trying to understand how the Zariski topology is different from the usual topology on the affine space $A^n$. Let $X$ be an affine algebraic subset of $A^n$, the $n$-dimensional affine space over $k$. Let $f_i:X\rightarrow A^1$, $i=1,...,k$ be continuous functions, both $X$ and $A^1$ being equipped with the Zariski topology. Is the function $f:X\rightarrow A^k, x\mapsto (f_1(x),...,f_k(x))$ continuous when $A^k$ is equipped with the Zariski topology? I know that $f$ is continuous for the product topology, but the Zariski topology is finer the the product topology.
I appreciate any help.