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Remarks concerning Darboux functions. (English) Zbl 0751.26004

The authors prove that: (1) there are a continuous function \(f:I\to R\) and a Darboux function \(g:I\to R\) such that \(fg\) has not the Darboux property; and (2) if \(f,g:I\to R\) are differentiable functions and \(f'\) or \(g'\) is continuous (\(f\) or \(g\) has no zero in \(I)\) then \(W(f,g)=fg'- f'g\) has the Darboux property.
Reviewer’s remarks: The result (2) is an immediate consequence of a theorem of A. M. Bruckner. The result (1) is well known. On page 6 of the cited monograph of A. M. Bruckner [Differentiation of real functions (1978; Zbl 0382.26002)] it is noted that there are a Darboux function \(f:I\to R\) and a continuous function \(g:I\to R\) such that \(f+g\) has not the Darboux property. Consequently, \(e^ f\) has the Darboux property, \(e^ g\) is continuous and \(e^ fe^ g=e^{f+g}\) has not the Darboux property.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21 Classification of real functions; Baire classification of sets and functions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems

Citations:

Zbl 0382.26002