Collect 'unusual' examples of real-valued functions

Question

I am trying to collect research papers, articles and books which contain 'weird' or 'unusual' examples of real-valued function.

An example of research paper is

A Counterexample and an Exact Versoin of Fatou's Lemma in Infinite Dimension

An example of article is

Some counterexamples on the behaviour of real-valued functions and their derivatives

Examples of books are

Surprises and Counterexamples in Real Function Theory

Analysis in examples and counterexamples. An introduction to the theory of real functions

Counterexamples in Analysis

Question: Can someone provide me more articles and books which are similar to the above? My aim is to learn as many function construction techniques as possible.

Answer 1

You may be interested in an example of a strictly positive real-valued function with zero Lebesgue integral in a model of Zermelo-Fraenkel set theory here.

Answer 2

A course of lectures entitled Some Pathological Functions in Real Analysis was held by A.B. Karazishvili in 1997 - 1998.

• His book Strange Functions in Real Analysis is based upon these lecture notes. Browsing the preface and the table of contents should give a first impression.

  Theorems, examples and exercises are often linked with corresponding references into the bibliography pointing to their origin.

This AMS review might be helpful.