The completeness condition is necessary for convergence to the exact solution. An incomplete element will generally converge to some other limiting function. The continuity condition is not necessary for convergence. The purpose of the continuity condition is to ensure that discontinuities at the interelement boundaries are not severe enough to introduce errors in addition to the discretization error.
If continuity and completeness conditions are satisfied, convergence is assured. An element which satisfies both conditions is called a conforming or compatible element. An element which satifies completeness but not continuity is called a nonconforming or incompatible element. For an expanded discussion of completeness and continuity see [45,34].
(See exercise 8.)