Perturbing the Mean Value Theorem: Implicit Functions, the Morse Lemma, and Beyond.

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. [ABSTRACT FROM AUTHOR]