Classification of discontinuities
The oscillation of a function at a point quantifies these discontinuities as follows: A special case is if the function diverges to infinity or minus infinity, in which case the oscillation is not defined (in the extended real numbers, this is a removable discontinuity).
For this kind of discontinuity: The one-sided limit from the negative direction:
The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point
[a] This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain.
For an essential discontinuity, at least one of the two one-sided limits does not exist in
(This is distinct from an essential singularity, which is often used when studying functions of complex variables).
Tom Apostol[3] follows partially the classification above by considering only removable and jump discontinuities.
His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem.
With the same purpose, Walter Rudin[4] and Karl R. Stromberg[5] study also removable and jump discontinuities by using different terminologies.
The term essential discontinuity has evidence of use in mathematical context as early as 1889.
[8] However, the earliest use of the term alongside a mathematical definition seems to have been given in the work by John Klippert.
[9] Therein, Klippert also classified essential discontinuities themselves by subdividing the set
is called an essential discontinuity of first kind.
is a bounded function, it is well-known of the importance of the set
In this theorem seems that all type of discontinuities have the same weight on the obstruction that a bounded function
are absolutely neutral in the regard of the Riemann integrability of
The main discontinuities for that purpose are the essential discontinuities of first kind and consequently the Lebesgue-Vitali theorem can be rewritten as follows: The case where
correspond to the following well-known classical complementary situations of Riemann integrability of a bounded function
One easily sees that those discontinuities are all removable.
belongs to one of the open intervals which were removed in the construction of
is an uncountable set with null Lebesgue measure, also
is a null Lebesgue measure set and so in the regard of Lebesgue-Vitali theorem
as before, we denote the set of all essential discontinuities of first kind of the function
Recall that Bolzano's theorem asserts that every continuous function satisfies the intermediate value property.
On the other hand, the converse is false: Darboux's theorem does not assume
to be continuous and the intermediate value property does not imply
Darboux's theorem does, however, have an immediate consequence on the type of discontinuities that
[11] This means in particular that the following two situations cannot occur: Furthermore, two other situations have to be excluded (see John Klippert[12]): Observe that whenever one of the conditions (i), (ii), (iii), or (iv) is fulfilled for some
On the other hand, a new type of discontinuity with respect to any function
is a bounded function, as in the assumptions of Lebesgue's theorem, we have for all
