# When Are There Continuous Choices for the Mean Value Abscissa?

## TL;DR

This paper investigates the conditions under which the mean value abscissa can be chosen as a continuous function of the interval's right endpoint, using advanced calculus tools like implicit function theorems and Morse's lemma.

## Contribution

It introduces simplified versions of the implicit function theorem and Morse's lemma to analyze the continuity of mean value abscissae, especially for analytic functions.

## Key findings

- For analytic functions, mean value abscissae can be chosen continuously with respect to the right endpoint.
- Develops simplified mathematical tools to analyze the continuity of solutions in calculus.
- Provides conditions under which the mean value abscissa varies continuously with the interval.

## Abstract

The mean value theorem of calculus states that, given a differentiable function $f$ on an interval $[a, b]$, there exists at least one mean value abscissa $c$ such that the slope of the tangent line at $c$ is equal to the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. In this article, we study how the choices of $c$ relate to varying the right endpoint $b$. In particular, we ask: When we can write $c$ as a continuous function of $b$ in some interval? Drawing inspiration from graphed examples, we first investigate this question by proving and using a simplified implicit function theorem. To handle certain edge cases, we then build on this analysis to prove and use a simplified Morse's lemma. Finally, further developing the tools proved so far, we conclude that if $f$ is analytic, then it is always possible to choose mean value abscissae so that $c$ is a continuous function of $b$, at least locally.