For $f : X \rightarrow Y$ a morphism of schemes, for $U \subset X$ a nonempty affine open, must $f(U)$ be contained in some affine open of $Y$?

Question

Is there a morphism of schemes $f : X \rightarrow Y$ admitting a nonempty affine open $U \subset X$ such that $f(U)$ is not contained in any affine open of $Y$?

Clearly if such $f$ and $U$ exist then $Y$ must be a non-affine scheme. However I'm not sure how to proceed further than this. Would anyone have any suggestions?

Answer 1

It depends what conditions you want to add... for instance take $\Bbb{P}^1$ over a field $k$ and note that there is a covering $\Bbb{A}^1_k \sqcup \mathrm{Spec}\:k \to \Bbb{P}^1$. Now, $\Bbb{A}^1_k \sqcup \mathrm{Spec}\: k = \mathrm{Spec}(k[x] \times k)$. The image under the obvious surjective map $\Bbb{A}^1_k \sqcup \mathrm{Spec}\:k\to \Bbb{P}^1$ is all of $\Bbb{P}^1$ and hence not contained in any affine open.