# Interpolating Between Choices for the Approximate Intermediate Value Theorem

## TL;DR

This paper proves the approximate intermediate value theorem constructively under weak hypotheses by interpolating between choices in a bisection-like algorithm, avoiding classical assumptions like uniform continuity and countable choice.

## Contribution

It introduces a novel constructive proof of the approximate intermediate value theorem that interpolates between classical bisection choices, requiring only pointwise continuity and weak hypotheses.

## Key findings

- Proves the approximate intermediate value theorem constructively.
- Develops an interpolation method between classical bisection choices.
- Avoids reliance on uniform continuity and countable choice.

## Abstract

This paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented with rational approximants, and without using countable choice. The theorem is that if a pointwise continuous function has both a negative and a positive value, then it has values arbitrarily close to 0. The proof builds on the usual classical proof by bisection, which repeatedly selects the left or right half of an interval; the algorithm here selects an interval of half the size in a continuous way, interpolating between those two possibilities.