Antiderivative - Antiderivatives of Non-continuous Functions

Antiderivatives of Non-continuous Functions

Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that:

• Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives.
• In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions are not Riemann integrable.

Assuming that the domains of the functions are open intervals:

• A necessary, but not sufficient, condition for a function f to have an antiderivative is that f have the intermediate value property. That is, if is a subinterval of the domain of f and d is any real number between f(a) and f(b), then f(c) = d for some c between a and b. To see this, let F be an antiderivative of f and consider the continuous function

on the closed interval . Then g must have either a maximum or minimum c in the open interval (a, b) and so

• The set of discontinuities of f must be a meagre set. This set must also be an F-sigma set (since the set of discontinuities of any function must be of this type). Moreover for any meagre F-sigma set, one can construct some function f having an antiderivative, which has the given set as its set of discontinuities.
• If f has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration.
• If f has an antiderivative F on a closed interval, then for any choice of partition, if one chooses sample points as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value F(b) − F(a).

However if f is unbounded, or if f is bounded but the set of discontinuities of f has positive Lebesgue measure, a different choice of sample points may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.