Semicontinuity properties of Kazhdan-Lusztig cells

29 pages. The paper arXiv:0805.3038 has been divided in two parts. This paper is an enriched (and translated) version of sections 1, 4, 5, 6 and 7 of arXiv:0805.3038.Computations in small Coxeter groups or dihedral groups suggest that the partition into Kazhdan-Lusztig cells with unequal parameters should obey to some semicontinuity phenomenon (as the parameters vary). The aim of this paper is to provide a rigorous theoretical background for supporting this intuition that will allow to state several precise conjectures