The Real And Complex Number Systems

Proof: Look at the Generalized Mean Value Theorem, we try to get something from
the equality.
fa fbg a 2
b ga gbf a 2
b , *
if fx, andgx satisfy following two conditions,
(i) f x gx and g x fx
and
(ii) fa fb f a 2
b ga gb g a 2
b ,
then we have the equality (*).
5.14 Given a function f defined and having a finite derivative f in the half-open
interval 0 x 1 and such that |f x| 1. Define a n f1/n for n 1, 2, 3, . . . , and
show that lim n a n exists.
Hint. Cauchy condition.
Proof: Consider n m, and by Mean Value Theorem,
|a n a m| |f1/n f1/m| |f p| 1 n m 1 1 n m
1
then a n is a Cauchy sequence since 1/n is a Cauchy sequence. Hence, we know that
lim n a n exists.
5.15 Assume that f has a finite derivative at each point of the open interval a, b.
Assume also that lim xc f x exists and is finite for some interior point c. Prove that the
value of this limit must be f c.
Proof: It can be proved by Exercise 5.16; we omit it.
5.16 Let f be continuous on a, b with a finite derivative f everywhere in a, b,
expect possibly at c. If lim xc f x exists and has the value A, show that f c must also
exist and has the value A.
Proof: Consider, for x c,
fx fc
x c f where x, c or c, x by Mean Value Theorem, *
since lim xc f x exists, given 0, there is a 0 such that as x c , c c,
we have
A f x A .
So, if we choose x c , c c in (*), we then have
fx fc
A x c f A .
That is, f c exists and equals A.
Remark: (1) Here is another proof by L-Hospital Rule. Since it is so obvious that we
omit the proof.
(2) We should be noted that Exercise 5.16 implies Exercise 5.15. Both methods
mentioned in Exercise 5.16 are suitable for Exercise 5.15.
5.17 Let f be continuous on 0, 1, f0 0, f x defined for each x in 0, 1. Prove
that if f is an increasing function on 0, 1, then so is too is the function g defined by the
equation gx fx/x.
Proof: Sincef is an increasing function on 0, 1, we know that, for any x 0, 1