I’m more of a novel writer where continuity is an easier, at least I feel, thing to achieve because it’s not a long-term thing like in TV shows where they write episodes one at a time. Either way, continuity is an important part of any media. It’s what ties everything together and lets a reader, or watcher, keep track of details. Once continuity is lost within a piece, it becomes confusing and the audience can sometimes be distracted, or even annoyed and angry, by the lack of it.
I watch a lot of television and continuity is a huge pet peeve of mine. When characters change ages suddenly or (and this is my biggest annoyance of all) state a specific desire and don’t follow through with it for narrative reasons, it makes me feel like the writers have gotten lazy. I absolutely hate when writers give characters something they want – most commonly, it’s going to college – and then fail to deliver. Continuity is thrown out the window in favor of “keeping the cast together” and they end up all, miraculously, going to school in the same town, at the same school, or not leaving home at all despite being smart, driven, and capable of going to college. I could name any number of TV shows where this has happened: Dawson’s Creek, Boy Meets World, The OC, Gossip Girl. Don’t make promises to readers/watchers that you don’t intend to keep.
Continuity, in that sense, is different than, say, the change of a flower between pages (looking at you, JK Rowling, Order of the Phoenix), but it’s still important. The devil is in the details. In actual publishing, there are continuity checkers whose job is to check that things are correct, but most writers don’t have that luxury, especially when starting out, so keep track of your details and keep your promises. Don’t have a character pine after Harvard and then have her go to Arizona State because her friends are there. It makes the readers feel cheated, unless there’s a damn good reason not to follow through, something that makes sense for the character and not for the writer.