The oscillation discontinuity and the discontinuity where the limit oscillation does not exist belong to the second kind of discontinuity. Note that there is no oscillation here, not infinite limit, so don't confuse it. Among the four kinds of discontinuity points in higher mathematics, oscillation discontinuity point is the most special and important discontinuity point, because oscillation is the only discontinuity point that may have indefinite integral (the existence theorem of original function) and the only discontinuity point of the second kind that may be integrated.

Oscillation discontinuity belongs to the second kind of discontinuity.

There is no doubt that any discontinuity x0 must be f(x0) does not exist (including defined nonexistence and undefined nonexistence) or exists but not on the function, that is, the value at the discontinuity x0 must not exist or exist and is not equal to the left and right limit of the point.

Generally, in Chinese mainland's textbooks, the discontinuity x0 can be undefined, but it is defined in the centripetal neighborhood of the discontinuity x0, that is, the discontinuity without definition on both sides will be discussed, and the discontinuity without definition on both sides will not be discussed, that is, what you have learned will not be discussed basically, and the discontinuity without definition on both sides will not be tested. Pay attention to this. However, in international textbooks, such as Fei's Calculus Course, there is a one-sided definition of discontinuity, that is, the same discontinuity can be infinite on the left and jumping discontinuity on the right.