Continuity errors can completely break the audience's connection with the material, so there are a few places to focus on to make sure that you avoid this kind of mistake.
Creating a film is more than just telling a story, it is about creating a whole world with identifiable rules and properties. Though this will usually resemble the real world, it is important to respect all of the events that occur with the proper principles of cause and effect. What this means is continuity, in all its forms, must be maintained if the audience is to suspend their disbelief and actually absolve themselves into the film they are watching. If you pay attention to certain parts of the film you can help to avoid continuity errors.
One of the most common continuity errors that occur in film are those around the passage of time. Often times things are not given the amount of time that is necessary for certain events to occur. For example, if you are constructing a story where a husband goes missing and then returns in two years to find his wife remarried with a one and a half year old child you have a major continuity error. This occurs on more minor scales all the time, where the drawing of an elaborate picture only takes a few moments or a car ride across a major metropolitan city is completed in less than five minutes. Respect the real laws of time and understand that people will pick up on this.
When people think about continuity issues they usually go directly to visual continuity within a scene or between adjoining scenes. This has to do with the way that the set looks, as well as the actors. If a character is dressed in a certain way in one angle then they must be identical in all other angels for that scene. Likewise, if the story requires that alterations are made in the scene by the actors then it needs to be seen in all subsequent views that the set has been altered. This means that if there is a fight in a scene and a living room is ripped apart then in later parts of that scene it must remain apparent that the alterations have been made. The best way to do this is to take clear stock of what the people look like in the beginning and end of each scene and what happens with their environment. If you are closing a day of filming before actually finishing a scene it is important to take photos of everything and everybody so they can be returned to their proper appearance once filming resumes.
Script organization is very important and this needs to be checked more than once. This is made much easier if you have a script supervisor and unit production manager. Events must remain linear in a script, unless there is a purpose for them not to. This does not mean that they have to be in the film in a certain order, but in the story space one event comes after another. Go through the script and make sure that not only are events in the correct order, but also to find any place where there is an event that has no correlated cause.
Attention to Detail
The most important thing is to make sure that everything is as it is intended to be, which requires inspecting every scene before shooting it. Not only will you be able to avoid continuity errors this way, but you will also be able to control the production more closely.