Let $X$ be an affine variety and $U\subset X$ Zariski open, Then, $U$ is dense in $X$ (i.e., the smallest set containing $U$ which is also the zero locus of a set of polynomials is $X$). Let $\varphi_1$ and $\varphi_2$ be continuous and agreeing on $U$.
Does it follow that $\varphi_1$ and $\varphi_2$ agree on $X$?
I'm trying to build intuition about how Zariski density is related to Euclidean density w.r.t. mappings.