Here's a funny essay on continuity in TV shows and the perils of trying to maintain continuity when shows cross over with each other; using the guide to TV crossovers as a resource, it demonstrates that almost any show can be connected to St. Elsewhere in some way, meaning that almost every show is part of the dream of that kid in the last episode.
What that essay got me wondering is when, exactly, episode-to-episode continuity became part of television shows. I'm not talking about big stuff, like the fact that if Lucy has a baby one season she's going to have it next season (though, starting the tradition that would be continued by Murphy Brown and others, we didn't see the kid very much after he was born); I mean the idea of letting the events of one episode influence, or restrict, later events on the show: we can't do a flashback to our hero's bar-mitzvah, because last year he said he was raised Catholic. (Another writer suggests: "Let's make him a Jewish Catholic. Problem solved.) As I've said elsewhere, TV shows with continuing characters used to aspire to the status and style of anthology shows -- which were considered, since the Playhouse 90 days, to be the best TV had to offer -- and so making every episode totally self-contained was not just convenient, but a mark of pride: we're creating self-contained stories here, not some soap opera. To have inter-episode continuity in, say, Star Trek would have been regarded as an unacceptable descent into soapiness.
That's why, as I've also said elsewhere -- I'm recycling a lot of my pop-culture talking points today, it seems -- the idea of continuity and "story arcs" often started on shows that didn't have particularly high aspirations, like Dallas. And the first show to have a lot of "continuity" in the sense that the above essay talks about -- that is, having characters remember what happened last week and letting that influence their actions -- may have been the gleefully lowbrow The Beverly Hillbillies. Paul Henning wrote or co-wrote every episode, and often he didn't really bother to create an ending for the episode; instead he'd just let the story peter out, and pick things up in the next episode. The upshot was that The Beverly Hillbillies, especially in the first couple of seasons, may have been the first true "arc" show. And, in general, there used to be more continuity on sitcoms than there was on dramas. Pre-Dallas dramas tended to stick to the idea of being anthology shows that happened to have continuing characters; but sitcoms in the '60s and '70s were more inclined to establish connections between the episodes. One of the writers of WKRP in Cincinnati kept a notebook in which he'd write down every point that was mentioned about a character -- what kind of car he or she drove, where he or she came from, etc. -- and he'd then refer to these points at story meetings, in order to keep them straight. He even wrote an entire episode to explain away various continuity discrepancies regarding one of the characters (Venus, who at various times was said to be a schoolteacher, a successful DJ, and a fugitive from justice). As dramas got more continuity-heavy, it seems, sitcoms got less so.
As to what I think of "continuity," I have to say that I am not turned off when a show makes a dreaded Continuity Error (tm). These things are always going to happen. Still, it is fun when a show tries to keep a character's background consistent, or incorporates references to earlier episodes, because it does increase the sense that you're watching people who could be real, and who do what real people do -- in this case, talk about things that previously happened to them.