The target of this chapter is to give some rudiments of(paraconsistent) real analysis, a dissection of the linear continuumas that which has no gaps. Axioms for the reals as a totally orderedcomplete field are given and developed, with philosophicalconsiderations about the nature of points. Focus is on the topologyof the real line, establishing the general principle at stake: thatif a change occurs, it must occur somewhere. This is confirmed atthe intermediate value theorem. Along the way, the (continuous)sorites paradox is “recaptured,” the existence ofinfinitesimal quantities is floated, and a theorem is proved about“splitting” geometric points.