An important tool to analyse the causal structure of a Lorentzian manifold is given by distance function. We define class functions on group contactomorphisms closed contact depending choice form. These are continuous with respect Hofer norm for defined Shelukhin [The contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if only orderable. To prove this, we show that intervals p...