Continuity sales are simply a contract to purchase products or services on an installment basis for a fixed period of time. That may sound complicated, but in practice, it actually is not. The best example of a continuity sale is a magazine subscription. The bigger the subscription, they better deal you get. The publisher gets more money up front, and the customer gets a better deal in the long run.
Continuity can apply to anything.
Let's say you own a dry cleaning business. Get people to pay your for the entire week up front for a lot of fast cash flow. You'll trade a discount for getting business, but you'll ensure a steady cash flow for months to come. Continuity works with just about any kind of product or service you are offering, from dry cleaning to to your personal consulting service. You can structure payments for continuity sales on almost any basis, but it's best by far to go for complete payment up front.