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arXiv · 1906.02026

When Are There Continuous Choices for the Mean Value Abscissa?

David Lowry-Duda·Miles H. Wheeler
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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.

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David Lowry-Duda, Miles H. Wheeler. 2019-06-05. When Are There Continuous Choices for the Mean Value Abscissa?. https://doi.org/10.1080/00029890.2021.1840879

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