We define surfaces as images of $C^1$ functions from $K \subset\mathbb{R}^2 \rightarrow \mathbb{R}^3$, with $K$ compact, and we say a surface is orientable is we can pick continuously a normal vector in each point. We don't have the tools of differential geometry.
Is there an elementary proof that the Moebius strip is not orientable? I'm looking for one that doesn't use a lot of...technology...maybe using path connectedness?
Thanks!