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Perturbing the mean value theorem: implicit functions, the Morse lemma, and beyond. (English) Zbl 1456.26006

One can begin with the authors’ 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 \((c, f (c))\) 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? As we explore this question, we touch on the implicit function theorem, a simplified version of the Morse lemma, and the theory of analytic functions.”
It is noted that the mean value theorem is one of the truly fundamental theorems of calculus. This theorem, auxiliary notions, and the main statements of the problem of the present research are briefly explained. Some examples are considered.
Several statements are proven with explanations. Some main results of this investigation are summarized in one theorem. Also, a special attention is given to further generalizations. It includes some explanations and proving certain statements.
Finally, some open questions related to the present research are given and discussed.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A06 One-variable calculus
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Software:

Matplotlib; SymPy; NumPy
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Full Text: DOI

References:

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