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arXiv · 2007.12752

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

Karol Gryszka·Paweł Pasteczka
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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.

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BibTeXRIS

Karol Gryszka, Paweł Pasteczka. 2020-07-24. On the $L^1$ and pointwise divergence of continuous functions. https://doi.org/10.1007/s10998-022-00500-7

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