Antiderivatives Exist without Integration

TL;DR

This paper provides a proof that continuous functions on closed intervals have antiderivatives without using traditional integral calculus, relying instead on Lebesgue's approach from 1905.

Contribution

It introduces a novel proof of the existence of antiderivatives that avoids classical integration methods, expanding theoretical understanding.

Findings

Antiderivatives exist for all continuous functions on closed intervals.

The proof does not rely on Riemann, Cauchy, or Darboux integrals.

Uses Lebesgue's 1905 approach to establish the result.

Abstract

We present a proof that any continuous function with domain including a closed interval yields an antiderivative of that function on that interval. This is done without the need of any integration comparable to that of Riemann, Cauchy, or Darboux. The proof is based on one given by Lebesgue in 1905.