Page 364 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Proof: Since f is differentiable, f is continuous. Define g (x) = [f(x + 1/n) – f(x)]/(1/n) and use [108].
n
Now we will show that a monotone real function (increasing or decreasing) is continuous and
differentiable at most of the points.

Let –¥  a < b  ¥ and let f: (a, b)   be a monotone function. Then, Y = {x  (a, b) : f is
discontinuous at x} is a countable set (possibly empty).
Proof: Suppose f is increasing. If x  Y, then necessarily f(x–) < f(x+), and we may choose a rational
number between f(x–) and f(x+). This gives a one-one map from Y to Q.
Definition: A collection  of non-degenerate intervals is a Vitali cover for a set X   if for each
 > 0, the subcollection {I  : 0 < |I| < } is also a cover for X.
[Vitali’s covering lemma] Let X   be such that m * L,1 [X] < ¥ and let  be a collection of intervals
forming a Vitali cover for X. Then,

(i) There are countably many pairwise disjoint intervals I, I ,…   such that m * [X\U I ] = 0.
1 2 L,1 n n
(ii) For every  > 0, there exist finitely many pairwise disjoint intervals I ,…,I   with the
1
k
property that m * L,1 [X\  k n 1 n
I ] < .
=
*
Proof: Write µ* = m L,1 for simplicity.
(i) With out loss of generality assume that every I   is a (non-degenerate) closed interval.
Choose an open set U   such that X  U and µ*[U] < ¥. Every x  X has a neighbourhood
contained in U. Hence  = {I  : I  U} is also a vitali cover for X. We will choose the
intervals I inductively. Let  = sup{|J|: J  } (note that  < µ*[U] < ¥) and let I   be
n 0 0 1
any interval with |I | >  /2. Suppose that we have chosen pairwise disjoint intervals
1 0
I ,…,I  . If X   n I , then we are done. Else, any x  X\  n I is at a positive distance
=
=
1 n i 1 i i 1 i
from the closed set  n I . Let  = sup{| J|: J   and I  J = Ø for 1  i  n}. Then 0 <  
=
i 1 i n i n
µ*[U] < ¥. Let I   be an interval with |I | >  /2. We will show that the sequence (I )
n + 1 n+1 n n
does the job.
Observation: For every J  , there is n   such that I  J  Ø (|I |  µ*[U] < ¥ so that
n n
(|I |)  0, and hence there is n   such that |I | < | J|/2).
n n
Let Y = X\  ¥ I and  > 0. We claim that µ*[Y] < . Let c be the midpoint of I and let Y   be
=
n 1 n n n n
the closed interval with midpoint c and |Y | = 6|I | (this Y may not be in ). Let k   be so
n n n n
I .
that å ¥ n k 1 |I | < /6. If x  Y, then in particular x does not belong to the closed set  k n 1 n
=
+
=
n
Choose J   with x  J and I  J = Ø for 1  n  k. By our observation above, I  J  Ø for some
n m
m  k + 1. Let m be the smallest such number. Then |J|   < 2|I | and hence |x – c |  |J| +
m–1 m m
|I |  3|I |. Therefore, x  Y . We have shown that Y   ¥ Y . Since å ¥ |Y |  6 å ¥
+
=
+
=
+
=
m m m n k 1 n n k 1 n n k 1
|I | < , we have proved that µ*[Y] < .
n
Now, note that the argument given above actually shows that for every  > 0, there is k   such
that µ* [X\  k I ] < . Hence we have established (ii) also.
=
n 1 n
When a mathematical problem is difficult, it is a good idea to divide the problem into many
subcases and to treat each case separately. If f : (a, b)   is a function, then the four Dini
derivatives of f at a point x  (a, b) are defined as follows.
f(x h) f(x)
+
-
D f(x) = limsup [upper right derivative]
+
h 0+ h
-
+
f(x h) f(x)
D f(x) = lim inf [lower right derivative]
+
0
h  + h
358 LOVELY PROFESSIONAL UNIVERSITY