Construction of Continuous, Integrable Functions with Extreme Behavior at Infinity

TL;DR

This paper constructs continuous, integrable functions exhibiting extreme behavior at infinity, where scaled versions of the function can have unbounded limsups, demonstrating novel properties of such functions.

Contribution

It introduces a method to build continuous, integrable functions with unbounded limsup behavior under scaling, expanding understanding of function behavior at infinity.

Findings

Constructed functions with unbounded limsup under scaling.

Demonstrated existence of continuous, integrable functions with extreme asymptotic behavior.

Extended the theory of function behavior at infinity.

Abstract

For any real sequence {c(n)} tending to infinity as n tends to infinity, this constructs a function f which is continuous and integrable, and such that for every nonzero x, limsup c(n) f(n x) is infinite.