What do you call a spike which does not flatten?
If I understand your question correctly, I would call the sudden spike to which you refer a jump, i.e., a sudden increase in the value of a function from some relatively narrow range of values it once had for some period of time to another relatively narrow range of much larger values that it continued to have for another (long) period of time. From Cambridge:
jump: to increase suddenly by a large amount, e.g., house prices have jumped dramatically; the cost of building the road has jumped by 70 percent.