§5.3 Continuity σ-Everywhere of p Given ρσ = 0

To explore the connection between differentials and deriva­ tives we shall need a continuity theorem for differentials. From Theorem 4 (§3.3) it is easy to conclude that for σ weakly archimedean and p tag-finite on a cell continuity σ-everywhere of p implies ρσ = 0. With stronger conditions on σ we can prove the converse. We need some definitions.