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.
