On the $L^1$ and pointwise divergence of continuous functions

Abstract

For a family of continuous functions $f_1,f_2,\dots :I\to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$I_f:=\{x\in I:\underset{n\to \infty }{\lim }{f}_n(x)=+\infty \}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$\underset{n\to \infty }{\lim }{\int }_x^y{f}_n(t)dt=+\infty \text{ for all }x,y\in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.

References in corpus (1)

• A Generalization of the Hausdorff Dimension Theorem for Fractals