“Continuance” is a term used in American legal English for the postponement or adjournment of legal proceedings. The proceedings continue when the continuance period comes to an end.
Confusion can arise when using the verb “continue”, which, depending on the context, can have two opposing meanings: either to resume proceedings after an interruption, as in “the trial continues tomorrow”; or to temporarily stop the proceedings for a specified period of time, as in “the judged ruled to continue the proceedings until after the summer break”. When used in the first sense, the sentence will usually indicate one defined time: e.g. “tomorrow”, “next week” “at 3 o’clock”, “in three weeks”, etc. When used in the second sense, it will indicate a period of time, often introduced by “until”. Thus, if the judge continues the proceedings until 4 September, the proceedings will continue on 4 September. If the judge continues the proceedings for three weeks, the proceedings will continue in three weeks (or three weeks later).
Since this use of continue and continuance is found only in American law, it is probably best to avoid it when writing in English yourself. Using “postpone” or “adjourn” instead will avoid confusion: “The judge postponed the hearing for three weeks and the proceedings continued at the end of the month”.