Continuous map example

Question

I am trying to understand the distinction between continuous maps between varieties and morphisms between varieties, and I believe a concrete example illustrating the distinction will help. What is an example of a continuous map $\pi:A\rightarrow B$ where $A,B$ are varieties and $\pi$ is not a morphism?

Answer 1

How about $A=B$ being the affine line over $\Bbb C$, and $\pi:z\mapsto\overline z$?

Answer 2

One easy source of examples is $\Bbb A^1_k$. Mapping the generic point to the generic point and permuting the closed points will produce a continuous map, and most examples of this fail to be morphisms. For instance, you can swap $0$ and $1$ and leave all other points invariant, which cannot be the result of a morphism.