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2017-10-10 Nonlinear Seminar

Simon Reinwand (University of Würzburg)
On Functions with Primitives

The problem of characterizing those functions, which possess a classical primitive has been investigated for decades. As far as we know, it is still an open problem if there is a “natural” characterization without involving integrals.

In this talk we give a quick overview about different characterizations while keeping our focus on probably the most natural one involving the Henstock-Kurzweil Integration Theory. While being a slight generalization of the Riemann Integral, it provides not only the most comprehensive version of the classical Fundamental Theorem of Calculus, but also yields a full characterization of the class of functions, which have a primitive.

We will discuss this class in more detail with respect to its size, its relation to other function classes, and some of its algebraic properties. Apart from recalling known and discussing new results we put a particular emphasis on examples and counter examples.

While answering questions regarding multiplication and change of variables we will point out connections between functions with primitives, Darboux-functions, HK-integrable functions and functions of bounded variation.