Vol. 83, No. 1, 1979

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Level sets of derivatives

R. P. Boas and Gerald Thomas Cargo

Vol. 83 (1979), No. 1, 37–44

We consider real-valued functions defined on intervals on the real line R, and we denote the extended real line by R.

The theme of this paper is the idea that, when a function has a derivative that is equal to some A R on a dense set, the derivative can take other (finite) values only on a rather thin set. Our most general result shows that, in particular, the hypothesis “the derivative is equal to A on a dense set” can be replaced by “at each point of a dense set, at least one Dini derivate equals A.” As corollaries we obtain unified and rather simple proofs of some more special known results, which we now state.

Mathematical Subject Classification 2000
Primary: 26A24
Received: 8 September 1978
Published: 1 July 1979
R. P. Boas
Gerald Thomas Cargo