chapter  6
Everywhere differentiable nowhere monotone functions
ByGilbert W. Bassett Jr., Roger Koenker
Pages 12

It follows from the results discussed in Chapter 4 that if a continuous function f : R → R $ f : \mathbf{R} \rightarrow \mathbf{R} $ is nowhere differentiable, then f is nowhere monotone, i.e., there does not exist a nondegenerate subinterval of R on which f is monotone. The present 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 the above-mentioned functions is obviously typical for classical mathematical analysis. In this connection, it should be noticed that many mathematicians of the end of the 19th century and of the beginning of the 20th century tried to give various constructions of functions of such a kind. As a rule, their constructions were either incorrect or, at least, incomplete. As pointed out in [104], the first explicit construction of such a function was suggested by Köpcke in 1889. Another example was given by Pereno in 1897 (this example is presented in [92]). In addition, Denjoy presented in his extensive work [58] a proof of the existence of an everywhere differentiable nowhere monotone function, as a consequence of his profound investigations concerning trigonometric series and their convergence. Afterwards, a number of distinct proofs of the existence of everywhere differentiable nowhere monotone functions were given by several authors (see, e.g., [79,104,156], [275]).