I have noticed that the CCSS do not include a single instance of the word “continuous”, and the standards do not highlight the continuous/discrete distinction. In a discussion on this blog, Jason Zimba here shows that the distinction is implicit in teaching number and measurement, and clearly there is opportunity to discuss the distinction while working with, e.g. slope in seventh grade (which is what I teach). But I would have expected to find these terms explicitly in the standards.
This implicit distinction also occurs earlier than Grade 7. As soon as students start working with the number line, they are beginning to move from a discrete counting model of the whole numbers to a continuous measurement model, which allows for the introduction of fractions. You make an interesting comment that this is not mentioned explicitly in the standards. One possible reason is that although the distinction is important background knowledge for teachers, it is not necessarily useful to introduce this terminology with students. For students, the related concepts are counting (discrete) and linear measurement (continuous). They should work with both of these, but it might not be worth while to introduce terminology, since it is difficult to see how the terminology would add to the intuitive conceptions, and the formalization of these conceptions is way beyond K–12.