A Remark on the Uniform Convergence of Some Sequences of Functions

Guy Degla

doi:10.4236/apm.2015.59048

Abstract

We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.

Highlights

Let X be a nonempty set, f : X → be a function and { }fn n∈ be a sequence of real-valued functions from X into
It is obvious that when X is finite and { }fn n∈ converges pointwise to f on X, { }fn n∈ converges uniformly to f on X
Let ε > 0 be arbitrarily fixed

Summary

Introduction

Let X be a nonempty set, f : X → be a function and { }fn n∈ be a sequence of real-valued functions from X into. It is obvious that when X is finite and { }fn n∈ converges pointwise to f on X, { }fn n∈ converges uniformly to f on X. If K is a compact metric space, f : K → a continuous function, and { }fn n∈ a monotone sequence of continuous functions from K into that converges pointwise to f on K, { }fn n∈ converges uniformly to f on K. Our aim is to highlight a new basic criterion that shows in some way how a sequence of real-valued functions can converge uniformly when it is more or less obvious that the sequence converges uniformly away from a finite number of points of the closure of its domain.

Theorem

Observation

Examples