Chapter 3.7, Problem 111E

To determine

(a)

To prove:

If $F\left(b\right)=f\text{'}\left(b\right)$ then F is continuous at $u=b$.

To determine

(b)

To prove:

if $x\ne a$, then $\forall u$

$\frac{f\left(u\right)-f\left(g\left(a\right)\right)}{x-a}=F\left(u\right)\frac{u-g\left(a\right)}{x-a}$.

To determine

(c)

To prove:

$\frac{f\left(g\left(x\right)\right)-f\left(g\left(a\right)\right)}{x-a}=F\left(g\left(x\right)\right)\frac{g\left(x\right)-g\left(a\right)}{x-a}$ and derive the chain rule by computing the limit of both sides as $x\to a$.