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 (Formula presented.), there exists at least one mean value abscissa c such that the slope of the tangent line at (Formula presented.) is equal to the slope of the secant line through (Formula presented.) and (Formula presented.). 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.