1. See Continuous function. (Mathematics.)
2. In film, continuity is consistency of the positions, colors, sizes, etc., of objects onscreen; continuity errors break the illusion of watching actual events. Care towards continuity must be taken because films are rarely, if ever, filmed in the order they are presented in: that is, a crew may film a scene from the end of a movie first, followed by one from the middle; the shooting schedule is sometimes dictated by weather, permitting issues, or other circumstances besides preference. Frequently film shoots will have a person dedicated exclusively to minding continuity.
One example of broken continuity occurs in the 1998 film Waking Ned Devine, when two of the characters are walking through a storm to Ned's house: the umbrella they are under is black during their conversation on the walk to the house (filmed from slightly above and to the front); yet after cutting to a lower shot from behind of Jackie approaching the house, Michael walks onscreen from the right holding an umbrella that is not black but beige, with a brown band at the rim.
Continuity also matters in other forms of art, such as novels and comics. If a character loses his jacket in one scene, yet he is still wearing it in a later scene without recovering it or getting another one, that is a break in continuity.
In a larger sense, continuity refers to consistent details of any fictional world, such as the names of characters' relatives.
Accidental or deliberate discrepancies in continuity are known as retconning, which stands for "retroactive continuity."