Fanon Discontinuity is the act of fans mentally writing out certain events in a show's continuity which don't sit well, no matter if it's a single episode, a season-length arc, an entire season or even an entire series. If a plot or ending rubs one the wrong way severely enough, fandom can just decide that the offending events never happened. On the series level, events may fall under Discontinuity because the show is perceived to suck at that point or decline too far in quality. Events also get "discontinued" for particularly screwing up the characters or setting, and a show that starts to suck will end up screwing things up eventually anyway.
In effect, Fanon Discontinuity is the opposite of Fanon (and not unrelated, either: a great amount of Fanon Discontinuity has resulted from violations of fanon). While extremely negative audience reactions may lead to an offending storyline being officially removed from canon in response, Fanon Discontinuity specifically refers to when fans disregard a storyline regardless of the creators' opinion on it.
In moviedom, Sequelitis is the most common cause of Fanon Discontinuity. It's very common to hear fans of a popular movie series disavow all sequels beyond a certain point, typically the first or second movie. For example, the unofficial slogan of the Highlander fandom is, "There Should Have Been Only One." Fanon Discontinuity also tends to arise when an audience has been dealt a particularly bad Wall Banger.
Sometimes discontinuity comes from not liking a very specific element while still enjoying everything else. When this happens, you've applied Broad Strokes to the Canon. Not to be confused with Negative Continuity.
One of the meta-causes of Alternate Universe.
If the questionable elements are written out of canon by the creators themselves, then said elements entered in Canon Discontinuity territory, or, luckily, are given a Discontinuity Nod. If the creator just bashes it, then it's Creator Backlash. If, on the other hand, the controversial element is somehow reworked into being tolerable or even popular, it's been Rescued from the Scrappy Heap. See also They Changed It, Now It Sucks.
NOTE: This is a very Subjective Trope, more based on the fandom rather than the event itself. The visceral response to Fanon Discontinuity can baffle other fans who don't take the event as seriously, or even *gasp* like the event. Please only post examples of the fandom as a whole disregarding an event. Also, using this as a Pothole is generally rather rude and irritating, so don't, unless you want to use Canon Discontinuity instead. If you would rather forget about it, then don't put it on the examples list!
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