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The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have $$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$ for at least one $c\in [a,b]$. We want to study the properties of such $r\in C_{f,a,b}:=\{c: c \text{ that satisfy (*)} \}=f^{-1}\big(\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)\big)$.

In our particular problem, we want to compare to it a different $c'$ from $$\frac{1}{\mu[0,a]}\int_{0}^{a}f(x)d\mu(x)=f(c')$$, by estimating their difference from below: $$c-c'>\delta.$$ (Our f is positive continuous but nondifferentiable and non-monotone and so we cannot simply invert).

Q1: How to express the condition $c-c'>\delta$ in terms of properties of f and $\mu$? What are some distinct requirements on $f,\mu$ for this lower bound to be true? Conversely, suppose that the MVT set is contained within a small neighbourhood $C_{f,a,b}\subset [x_{0}-\epsilon,x_{0}+\epsilon]$. What does that imply for the behaviour of f (at least when $\mu=$Lebesgue)?

Q2: But I am curious if there is a framework for studying the properties of $C_{f,a,b}$? Is there any literature out there on putting bounds on the inf and sup of this set for various types of functions?

Q3: A geometric interpretation for such c is as the values that give matchings area in (*). I am curious if there have been any more theoretical results using that idea. Maybe studying some minimization problem: $$arginf\{|\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)-f(r)|^{2}\}$$ using some optimization algorithm. enter image description here (from MVT's wiki)

  1. If $|c-c'|\leq \delta$ for all $c\in C_{f,a,b},c'\in C_{f,0,a}$ then by the geometric interpretation and the Second-MVT, we intuitively see that the function must be like a trapezoid reaching a plateau inside the $\delta-$neighbourhood of $x=a$. Inside that plateau f must have a few high peaks that have enough area underneath them to give a mean value equality. This is assuming we work with $\mu$ Lebesgue; for more general measures, we can get different shapes.
  2. Some work on exploring this same reverse question: given c what can we say about f: "On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions" and "When Are There Continuous Choices for the Mean Value Abscissa?".
  3. Partial Converse to the mean value theorem
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