According to Wikipedia, continuity is "consistency of the characteristics of persons, plot, objects, places and events seen by the reader or viewer" of fictional works.
The term may be applied in the Grand Theft Auto series to compare the consistency in a number of fictional topics in each game. It is difficult to ascertain whether continuity errors in many of the games are intentional or not, as settings and characters have been frequently redesigned, while plot lines in certain games are known not to relate to each other. It can be assumed Rockstar North, the series' primary developer, has little regards to continuity when the need arises.
A degree of effort was made to maintain continuity in the 3D Universe, with many characters recurring between titles, the settings maintaining their designs, and Grand Theft Auto III's plot expanded. Minor continuity errors, however, were still persistent in this part of the series. Similarly, continuity in HD Universe remain, although it no longer link's to the previous games.
One of the most glaring lack of continuity is the design of cities in the series. The three cities of the original Grand Theft Auto, Liberty City, San Andreas and Vice City, would go on to be featured in future games, but were significantly redesigned, with San Andreas redone as a state with three cities. Liberty City, in particular, has the highest total with three iterations, barring another two adapted for handheld console versions. Other locales, such as London and Anywhere City, were not featured again after their first appearances.
For games taking place in the 3D Universe, their settings maintained a degree of consistency in layout and appearance when featured again in another game, although all of them would be set years before GTA III and thus would also receive changes in certain locations of the settings. Grand Theft Auto: San Andreas would even feature a segment of Liberty City featured in GTA III, confirming the city's existence with San Andreas. All in all, the games' iteration of Liberty City has been featured three times (four if GTA: San Andreas is counted), followed by Vice City with two appearances, and San Andreas with only one appearance.
For much of the series, many characters in the series were largely disposable, appearing only once in a game. This was most apparent in GTA 1, GTA London and GTA 2, where the many bosses the player works for, will never appear again in succeeding games. However, certain elements of past characters have been reused, as is the case with El Burro and Claude Speed.
The only line of games to extensively reuse characters are games in the 3D Universe, where characters introduced in a game would reappear in an older or younger form in games released after. Despite the attempted continuity in the games, several minor continuity errors may be noted, such as their appearance, mannerisms and personality.
Grand Theft Auto IV started anew by dropping all characters featured in previous games, creating the HD Universe, while establishing a new cast of characters intended to appear in the same timeline in the base game, and the installments that follow. Exceptions to the case are recurring personalities of past radio stations. Lazlow, who has extensively appeared on radio in the 3D Universe, is present in Integrity 2.0 and Chattersphere, the game alluding to Chatterbox FM, which Lazlow hosted in GTA III and Grand Theft Auto: Liberty City Stories. Likewise, Fernando Martinez cameos in The Ballad of Gay Tony in the form of a radio DJ for Vice City FM, and also Cliff Lane and Andee returning in Grand Theft Auto V as DJs in the radio station Self Radio (PC version only).
In the 2D Universe, plot lines were not made to interconnect each other. As characters and settings differ between the games, it is assumed the games each feature their own distinct storyline and thus were not related.
In the 3D Universe, the original plot of GTA III was expanded towards the past, providing backgrounds on various events and characters, with the universe coming to a close in Grand Theft Auto: Vice City Stories.
The HD Universe, however, does not reference events in previous games, setting themselves apart from their predecessors, except for certain easter eggs; like graffiti inside apartment buildings saying that every protagonist from the 3D Universe had died, or Lazlow mentioning his time on Chatterbox FM.