ABSTRACT

In 1884 the Swedish mathematician Mittag-Leffler discovered that for every domain G in the complex plane C there is at least a holomorphic function f : G → C that is not continuable as a holomorphic function to any larger domain. The study of the lineability of this interesting phenomenon will be the first subject of this chapter. Special attention will be paid to the case G = D := {z ∈ C : |z| < 1}, the open unit disc. Besides, there are a few other kinds of strange functions, such as strongly annular functions, entire functions tending to zero on every straight line but with a speedy overall growth, functions having a wild behavior near the boundary of the domain and infinitely differentiable complex functions on a real interval which are nowhere almost analytic in the sense of Gevrey. All of them will be investigated in this chapter from the point of view of lineability.