On the level structure of bounded derivatives. (English) Zbl 1064.26003

Cater, F. S.

Real Anal. Exch. 29(2003-2004), No. 2, 657-662 (2004).

Summary: We prove: In the space \({\mathcal C}\) of continuous functions on \([0,1]\) under the sup metric, the functions all of whose level sets (in every direction) have measure zero, form a residual subset of \({\mathcal C}\). In the space \({\mathcal D}\) of bounded derivatives on \([0,1]\), the derivatives all of whose level sets are nowhere dense sets of measure zero form a residual subset of \({\mathcal D}\). Moreover, there exists a derivative in \({\mathcal D}\) all of whose level sets have measure zero and one of whose level sets is dense in \([0,1]\).

MSC:

26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A12	Rate of growth of functions, orders of infinity, slowly varying functions

Keywords:

derivative; level set; measure; category; graph; residual subset