These words can mean the same thing or different things depending on the context in which they are used.
Continuation - the act or fact of continuing in or the prolongation of a state or activity or resumption after an interruption or something that continues increases, or adds.
Continuance - can be the same as continuation or the extent of continuing (duration) or the quality of enduring (permanence) or adjournment of a court case to a future day.
Continuity - uninterrupted connection, succession, or union or uninterrupted duration or continuation especially without essential change or something that has, exhibits, or provides continuity: as a: a script or scenario in the performing arts b: transitional spoken or musical matter especially for a radio or television program c: the story and dialog of a comic strip or the property of being mathematically continuous
With help from Webster ;-)