Mean value theorem for Riemann-integrable functions

Question

I'm reading Bressoud's A radical approach to Lebesgue theory of integration and there's a section that I don't get, please read below:

Is there a mean value theorem for integrable functions ? I know there's one for integrals of continuous functions. If the continuity assumption is dropped, I don't know what to do...

Do you mean the Average Value Theorem, i.e. the existence of a $c \in (a, b) $ for which $$ \frac{1}{b-a} \int_a ^b f(x) \ dx = f(c) $$ provided $f$ is continuous? — Sean Roberson, Aug 18 '17 at 17:23
https://math.stackexchange.com/questions/429769/is-an-integral-always-continuous — platty, Aug 18 '17 at 17:23
@LeGrandDODOM the result shows that $F$ is continuous, so you can apply the MVT — platty, Aug 18 '17 at 17:27
@platty For the MVT on $F$, you would need differentiability. — Fimpellizzeri, Aug 18 '17 at 17:32
Taking a second look, I think the claim is false without further assumptions. Let $f = 1$ from $0$ to $1$ and $f = 2$ from $1$ to $2$. Taking $a=0,h=2$ implies that there is some $c \ in (0,2)$ with $f(c) = \frac{3}{2}$, unless I've fudged the numbers somewhere. — platty, Aug 18 '17 at 17:37

RRL · Accepted Answer · 2017-08-18T18:49:15.457

4

Suppose $f$ is integrable (not necessarily continuous) and $\lim_{x \to a+} f(x)= L$. For any $\epsilon > 0$ there is a $\delta > 0$ such that $|f(x) - L| < \epsilon$ when $0 < x < a + \delta$.

If $0 < h < \delta$, then

$$\left|\frac{1}{h}\int_a^{a+h} f(x) \, dx - L \right| = \left|\frac{1}{h}\int_a^{a+h} (f(x) - L) \, dx \right| \leqslant\frac{1}{h}\int_a^{a+h} |f(x) - L| \, dx < \epsilon $$

Thus, $\displaystyle \lim_{h \to 0+} \frac{1}{h}\int_a^{a+h} f(x) \, dx = L$.

A similar argument applies to the left-hand limit. The mean value theorem for integrals is not needed (nor does it apply to discontinuous functions.)

edited Aug 18 '17 at 18:49

answered Aug 18 '17 at 18:39

RRL

90,707

Yes, there's a mistake in the book, as I suspected. – Gabriel Romon Aug 18 '17 at 18:57
This is another mistake is Bressoud's book (the first one which you discovered was regarding proof of equivalence of Cauchy integrability and Riemann integrability). +1 – Paramanand Singh Aug 19 '17 at 05:04
Btw this is the way one proves that the derivative of $\int_{a} ^{x} f(t) , dt $ at $c$ is $f(c) $ provided $f$ is continuous at $c$. Most textbooks use mean value theorem to prove this, but then it applies only if $f$ is continuous not just at $c$ but in some neighborhood of $c$. – Paramanand Singh Aug 19 '17 at 05:07
@ParamanandSingh: Thanks. It really is quite a nice book, particularly the historical perspective it provides and the later sections on Lebesgue's FTC. – RRL Aug 19 '17 at 05:11
I have the same opinion about Bressoud's book, but I guess these mistakes were created by the burden of writing a book. Long back I tried to write a book on analysis (not exactly for publishing) but dropped it just after writing 120 pages of $\mathrm \LaTeX $ and the paucity of nice exercises (MSE was yet to born otherwise there would have been no paucity of exercises) – Paramanand Singh Aug 19 '17 at 05:15

Mean value theorem for Riemann-integrable functions

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