ABSTRACT
This chapter is devoted to some constructions of functions also acting from R into R, differentiable everywhere but nowhere monotone. The question of the existence of such functions is obviously typical for classical mathematical analysis. And it should be noticed that many mathematicians of the end of the 19th century and of the beginning of the 20th century tried to present various constructions of the above-mentioned functions. As a rule, their constructions were either incorrect or, at least, incomplete. As pointed out in [83], the first explicit construction of such a function was given by Ko¨pcke in 1889. Another example was suggested by Pereno in 1897 (this example is presented in [74], pp. 412-421). In addition, Denjoy gave in his extensive work [48] a proof of the existence of an everywhere differentiable nowhere monotone function, as a consequence of his deep investigations concerning trigonometric series and their convergence.
