A misleading example of a test function in space D from textbooks on Generalized Functions and Distributions

TL;DR

The paper highlights a common textbook error where a supposed infinitely differentiable bump function in space D is actually not infinitely differentiable, emphasizing the need for correction in mathematical education.

Contribution

It identifies and clarifies a widespread misconception in textbooks regarding the properties of bump functions in space D.

Findings

The example in textbooks is mathematically incorrect.

The function labeled as a bump function is not infinitely differentiable.

This error affects the understanding of generalized functions.

Abstract

In many textbooks on Generalized Functions and Distributions, we can find an example of infinitely differentiable function of bounded support from space D, called a bump function. This example is incorrect since this function is not infinitely differentiable.