Mapping Diagram of a Function
$f: RP^1 \rightarrow RP^1$

The mapping diagram of a function is the set of arrows from elements of $RP^1$ to elements of $RP^1$ .
We consider $RP^1$ to be $S^1$, the circle, so the mapping diagram is a ruled "surface" with one boundary $S^1$ and the other a subset of  $S^1$.

MD xMD x^2-4MD 1/(x^2-1)