Analysis of Integral of a continuous function one more question today I've been thinking on...
Prove that if $f$ is continuous on $[a,b]$, $0<a<b$, then $\int_{a}^{b} {f(t) \over t} dt$ $=$ $\int_{a}^{s} {f(t) \over a} dt$ for some $s \in [a,b]$. 
Intuitively this seems clear. $f$ being continuous means that if we view the $f(t)/a$ integral from $a$ to $b$ as a limit of its integral from $a$ to $s$ as $s \to b$, then we should find some intermediate $s$ that "catches" the $f(t)/t$ integral as $s$ slides along the integral since $1/t\leq1/a$. Problem is I can't find a nice way to write it formally. Thoughts?
EDIT: I was able to prove the following:
If $f$ is integrable on $[a,b]$ and (WLOG) nonnegative, and $g$ is monotone decreasing and positive, then we have an $s$ in $[a,b]$ such that $\int_{a}^{b} fg = g(a)\int_{a}^{s} f$, which happens to be exactly the conditions we need on the problem above. I believe it should be true for arbitrary (even non constant) sign on $f$, and $g$ shouldn't have to be decreasing or positive, but it's something, and it solves the problem. Not to mention it is easy to prove with a simple application of IVT.
 A: I found a complete proof: Here. This is the 3rd situation. This proof doesn't involve measure theory.
A: I have been working on this problem recently, too! In order to show it I think it helps first to prove the following:

Theorem
If $f$ is continuous and $g$ is integrable and non-negative, then there is some $\xi \in [a,b]$ such that $$\int_{a}^{b} f(x)g(x) \, dx = f(\xi)\int_{a}^{b} g(x) \, dx$$

This you can prove in the natural way you described (by thinking about the maximum and minimum of $f,$ I'll leave you the joy of doing that one!)
Now to your problem. We use integration by parts and the above theorem to proceed. Define $F = \int_{a}^{x}f(t)\,dt$ to write
$$ \int_{a}^{b} \frac{f(t)}{t} dt = \left[\frac{F(t)}{t}\right]_{a}^{b} + \int_{a}^{b}\frac{F(t)}{t^{2}} dt = \frac{F(b)}{b} + F(\xi)\int_{a}^{b}\frac{1}{t^{2}}dt$$
for some $\xi \in [a,b]$ (by the above theorem). If you work this out, you get to an answer very similar to yours but a little different. I wonder, was your answer a conjecture or have you been asked to show this by a lecturer/book?
EDIT:
I think the conclusion should be
$$ \exists \xi \in [a,b]: \int_{a}^{b} f(x)g(x) = g(a)\int_{a}^{\xi}f(x)\,dx + g(b) \int_{\xi}^{b} f(x)\,dx $$
and not simply
$$ \exists \xi \in [a,b]: \int_{a}^{b} f(x)g(x) = g(a)\int_{a}^{\xi}f(x)\,dx$$
Otherwise we could choose, for example, $g(x) = x, f(x) = 1$ and $[a,b] = [0,1].$ Similar considerations lead you to conclude that $g$ must be either non-decreasing or non-increasing (but it doesn't matter for $f$). The integration by parts method outlined above for $g(t) = 1/t$ should work as a proof for any such $g$ with $g^{\prime}$ continuous.
