Question about proof of the fundamental theorem of calculus I am reading the book Advanced Calculus by Edwards, H. M. (1980), and I came across the following proof of the fundamental theorem of calculus, which is given as below.


I do not understand the following parts of the proof.


*

*It says that the expression (12) "can be thought of as the average difference per unit length between...".
What does it mean by "the average difference per unit length"?
I understand that the "per unit length" part comes from $\frac{1}{\Delta t}$ in (12), what about the "average difference" part?  

*How does $|\epsilon_{ab}| \geq \frac{1}{2}(|\epsilon_{ac}|+|\epsilon_{cb}|)$ imply that $|\epsilon_{ac}| \geq |\epsilon_{ab}|$ or $|\epsilon_{ac}| \geq |\epsilon_{ab}|$
 A: *

*The difference between the integral and the other expression, taken over the whole interval from $a$ to $b$, is $$\left[\int_a^b F'(t)\,dt\right]-\Delta F(t)$$ If we divide by the length of the interval $\Delta t=b-a$, then we get the total difference per length, or the average of the difference over the interval. For instance, a difference of $10$ over an interval of length $5$ is an average difference of $2$ per unit length.

*Suppose that both $|\epsilon_{ac}|<|\epsilon_{ab}|$ and $|\epsilon_{cb}|<|\epsilon_{ab}|$. Then $${1\over 2}\left(|\epsilon_{ac}|+|\epsilon_{cb}|\right)<{1\over 2}\left(|\epsilon_{ab}|+|\epsilon_{ab}|\right)=|\epsilon_{ab}|$$ a contradiction, as the two are equal.

I should add that the wording of the proof is a bit misleading. The proof does not in fact analyze the errors over each of the sub-intervals into which $[a,b]$ is partitioned to take the Riemann Sum. Instead, it only analyzes one error - namely, that between the integral and the expression that the integral is claimed to equal - and shows that this one error is less than a quantity which tends to zero.
