As we know Continuation is used for asynchronous callouts to make long-running requests from a Visualforce page to an external Web service and process responses in callback methods. Asynchronous callouts that are made from a Visualforce page don’t count toward the Apex limit of 10 synchronous requests that last longer than five seconds.

An asynchronous callout should made from a Visualforce page and the response is returned through a callback method. So you cannot use a method that is not called from Visualforce page, but we can use action method to call Continuation callout.

When repeating action need to perform then action function is only way for asynchronous callout of Continuation.