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umm that reference has been published outside the arXiv
Matthew Frank - Interpolating Between Choices for the Approximate Intermediate Value Theorem lmcs:2638 - Logical Methods in Computer Science, July 14, 2020, Volume 16, Issue 3 - https://doi.org/10.23638/LMCS-16(3:5)2020
Logical Methods in Computer Science, July 14, 2020, Volume 16, Issue 3 - doi:10.23638/LMCS-16(3:5)2020
This paper by Auke Booij gives another proof of the intermediate value theorem without the axiom of choice by using locators and locally non-constant functions.
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