early 15c., "uninterrupted connection of parts in space or time," from Old French continuité, from Latin continuitatem (nominative continuitas) "a connected series," from continuus "joining, connecting with something; following one after another," from continere (intransitive) "to be uninterrupted," literally "to hang together" (see contain).
Cinematographic sense, in reference to assuring there are no discrepancies of detail in linked scenes filmed at different times, is recorded by 1919, American English. It was originally especially women's work.
The scenario,--that is the division of the synopsis into scenes from which the picture is made--is written by men and women specially trained for the work. Women are as successful, perhaps more so, in this line than men. The average price for an original motion picture synopsis is from $500 to $1500, but the price may be higher or lower according to the company and the value of the author's name. ... Continuity writers or those who divide the story into the scenes (continuity and scenario being different names for the same thing) are specially well paid. [Helen Christene Hoerle and Florence B. Saltzberg, "The Girl and the Job," New York, 1919]