Continuous change refers to a kind of variation over time that is expected and understood.
The gradual growth of a child into an adult is continuous change. Though such growth may involve times of confusion and upheaval, nothing about human growth is new or unexpected.
This is the kind of change the Church in North America is familiar with. As Alan Roxburgh stated in the above quote, the training of new pastors and leaders has for a long time been based on the assumption of slow and steady variation over time.
Discontinuous change is different. Discontinuous change is not understood because there is no precedent for this kind of change.
For instance, whereas the growth from childhood to adulthood is a form of continuous change, an example of discontinuous change would be if the child were to lose both parents and become an orphan.
A disturbing example, I know. But it illustrates the severity of such an unanticipated development. It's completely new. It's unexpected, difficult to understand, and requires us to respond with skills we don't yet have.