## Real Analysis Exchange

### On the level structure of bounded derivatives.

F. S. Cater

#### Abstract

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 of $[0,1]$, 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 $[0,1]$.

#### Article information

Source
Real Anal. Exchange, Volume 29, Number 2 (2003), 657-662.

Dates
First available in Project Euclid: 7 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149698556

Mathematical Reviews number (MathSciNet)
MR2083804

Zentralblatt MATH identifier
1064.26003

#### Citation

Cater, F. S. On the level structure of bounded derivatives. Real Anal. Exchange 29 (2003), no. 2, 657--662. https://projecteuclid.org/euclid.rae/1149698556

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