TITLE: Collect 'unusual' examples of real-valued functions
QUESTION [8 upvotes]: 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.

REPLY [6 votes]: 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.