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Remarks on the infinite-dimensional counterparts of the Darboux theorem
  • Wojciech Kryszewski,
  • Piotr Juszczyk
Wojciech Kryszewski
Lodz University of Technology Faculty of Technical Physics Information Technology and Applied Mathematics

Corresponding Author:wojciech.kryszewski@p.lodz.pl

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Piotr Juszczyk
Lodz University of Technology Faculty of Technical Physics Information Technology and Applied Mathematics
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Abstract

The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, i.e. attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux’s theorem when the derivative is continuous, states that a continuous real-valued function $f$ defined on $[-1,1]$ satisfying $f(-1)<0$ and $f(1)>0$ 0, has a zero, i.e. $f(x) = 0$ for at least one number $-1
13 Jun 2022Submitted to Mathematical Methods in the Applied Sciences
14 Jun 2022Submission Checks Completed
14 Jun 2022Assigned to Editor
20 Jun 2022Reviewer(s) Assigned
09 Sep 2022Review(s) Completed, Editorial Evaluation Pending
26 Sep 2022Editorial Decision: Revise Minor
05 Oct 20221st Revision Received
06 Oct 2022Submission Checks Completed
06 Oct 2022Assigned to Editor
17 Oct 2022Review(s) Completed, Editorial Evaluation Pending
18 Oct 2022Editorial Decision: Accept