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

TL;DR

This paper investigates the divergence properties of increasing sequences of continuous functions, focusing on the sets where they tend to infinity and their relation to the divergence of their integrals, motivated by quasiarithmetic means.

Contribution

It characterizes the structure of sets where continuous functions diverge and explores their properties under integral divergence conditions.

Findings

Characterizes possible divergence sets for increasing continuous functions.

Analyzes the relationship between pointwise divergence and integral divergence.

Provides insights into the limit behavior of quasiarithmetic means.

Abstract

For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ 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 $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.