Equipartition of a segment

TL;DR

This paper proves that any segment can be divided into m parts with equal values of a continuous function, even if the function takes positive and negative values, including degenerate segments.

Contribution

It introduces a general proof for partitioning segments into equal-function-value parts under broad conditions, extending previous results to functions with negative values.

Findings

Any segment can be partitioned into m parts with equal function values.

The function can take positive and negative values, including zero on degenerate segments.

The partitioning holds for any positive integer m.

Abstract

We prove that, for any positive integer $m$, a segment may be partitioned into $m$ possibly degenerate or empty segments with equal values of a continuous function $f$ of a segment, assuming that $f$ may take positive and negative values, but its value on degenerate or empty segments is zero.