Tag Archives: continuous

Continuity of the Mean Value

1. Introduction

When I first learned the mean value theorem as a high schooler, I was thoroughly unimpressed. Part of this was because it’s just like Rolle’s Theorem, which feels obvious. But I think the greater part is because I thought it was useless. And I continued to think it was useless until I began my first proof-oriented treatment of calculus as a second year at Georgia Tech. Somehow, in the interceding years, I learned to value intuition and simple statements.

I have since completely changed my view on the mean value theorem. I now consider essentially all of one variable calculus to be the Mean Value Theorem, perhaps in various forms or disguises. In my earlier note An Intuitive Introduction to Calculus, we state and prove the Mean Value Theorem, and then show that we can prove the Fundamental Theorem of Calculus with the Mean Value Theorem and the Intermediate Value Theorem (which also felt silly to me as a high schooler, but which is not silly).

In this brief note, I want to consider one small aspect of the Mean Value Theorem: can the “mean value” be chosen continuously as a function of the endpoints? To state this more clearly, first recall the theorem:

Suppose $latex {f}$ is a differentiable real-valued function on an interval $latex {[a,b]}$. Then there exists a point $latex {c}$ between $latex {a}$ and $latex {b}$ such that $$ \frac{f(b) – f(a)}{b – a} = f'(c), \tag{1}$$
which is to say that there is a point where the slope of $latex {f}$ is the same as the average slope from $latex {a}$ to $latex {b}$.

What if we allow the interval to vary? Suppose we are interested in a differentiable function $latex {f}$ on intervals of the form $latex {[0,b]}$, and we let $latex {b}$ vary. Then for each choice of $latex {b}$, the mean value theorem tells us that there exists $latex {c_b}$ such that $$ \frac{f(b) – f(0)}{b} = f'(c_b). $$
Then the question we consider today is, as a function of $latex {b}$, can $latex {c_b}$ be chosen continuously? We will see that we cannot, and we’ll see explicit counterexamples. This, after the fold.

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