  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.
