With regard to the OT and the NT, no one argues for strict continuity or strict discontinuity. That is, everyone believes in some continuity (e.g., we believe in one God) and some discontinuity (e.g., we no longer offer animal sacrifices). The crux of the problem is determining how much stays the same (continuity) and how much is different (discontinuity).
Dispensationalists interpret the Scriptures through the grid of presumed discontinuity. This is their main hermeneutical presupposition. The NT is radically different from the OT. Unless something is explicitly repeated in the NT, then it must not be valid for today. Such an assumption results in conclusions that affect virtually every area of theology.
One of the most painful memories for Dispensationalists is this unfortunate statement in the 1909 Scofield Reference Bible:
"As a dispensation grace begins with the death and resurrection of Christ (Rom. 3:24-26; 4:24, 25). The point of testing is no longer legal obedience as the condition of salvation, but acceptance or rejection of Christ, with good works as the fruit of salvation."
This statement has been rightly rejected by later Dispensationalists, but one has to ask, how could Scofield have even thought that?
The blame should be placed on the hermeneutical grid of presumed discontinuity. Early Dispensationalists were committed to radical discontinuity. They saw discontinuity everywhere. The task of the interpreter was to learn how to "rightly divide the word of God," that is, to separate old from new.
As Dispensationalism developed, there has been a conscious move away from radical discontinuity. Only a Classical Dispensationalist could have argued for radical discontinuity in soteriology. Revised Dispensationalism rejected all such notions.
Yet, Revised Dispensationalists continued to hold to radical discontinuity in other areas, such as different eternal destinies for Israel and for the Church (Israel on earth and the church in heaven). Progressive Dispensationalists have rejected some of these most notorious peculiarities, but they are still operating under the assumption of presumed discontinuity.
As long is there is presumed discontinuity, there will be Dispensationalism. It is likely that there will be another stage of development after Progressive Dispensationalism, which will be another movement towards continuity while stubbornly attempting to hold to presumed discontinuity.
At some point, Dispensationalists should question their commitment to presumed discontinuity. Try the alternative and see if it doesn’t fit the Scriptures better.