Some new simple proofs of old difficult theorems
First, the author discusses the problem of the existence of a nowhere monotonic everywhere differentiable function and gives some very interesting new short constructions of such functions. Some recent results in the theory of the differentiation are the sources for the ideas of these constructions. Then follows a new proof of the existence of a differentiable function with dense sets of strict maxima and strict minima. This problem is related with the previous one and his proof is an application of a theorem of Fleissner-Foran appeared in 1976. The last part of the paper deals with the following Saks theorem: The typical continuous function has infinite unilateral derivatives on sets of the cardinality of the continuum. There is published the new short proof of the Saks theorem communicated by M. Laczkovich who attributed it to D. Preiss. Namely, the Saks theorem is an easy consequence of the following theorem published by V. Jarník in 1934: Typical continuous functions are nowhere approximately differentiable.
nowhere monotonic everywhere differentiable function
scientific article; zbMATH DE number 3908746