Page 104 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 
 |n|y – x| + n|x – x| +
j
4
2n 
 +
k 4


2
This is a contradiction. So f  S .
n
 If f(x) exists find  > 0 s.t. " 0 < |y – x| <,
f(y) f(x) f(y) f(x)
-
-

- f (x) < < 1 |f (x)|
+

-
-
y x y x
f(y) f(x)
-
Function y  is continuous on [0, 1]\(x – , x + ) which is compact. Hence the
y x
-
function is bounded, so  n   s.t.
-
f(y) f(x)

y [0, 1]\(x -  , x +    n
)
-
y x
May take n > 1 + |f(x)| so get inequality holding " y  [0, 1]\{x}.
Then |f(y) – f(x)|  n|y – x| " y  [0, 1]. (This clearly holds for y = x and holds by the above
for y  x.) So if  f  C[0, 1] s.t. f(x) exists for some x then f  S .
n
 These three parts together complete the proof, since by Baire (5.17) C[0, 1] is not meagre,
so there must be a function which is not differentiable at any point, as any that are
differentiable at at least one point are in a nowhere dense subset.
8.8 Keywords
Complete Metric: Subspace C of complete metric M compact iff closed and totally bounded.
Cantor: Let F decreasing sequence of non-empty closed subsets of metric M s.t. diam
n
(F ) ¾¾¾® 0. Then  ¥ F  .
0 /
=
n n®¥ n 1 n
Continue Inductively: Having defined x ,  use fact that G dense to find x  G  B
k–i k–i k k k
æ  k 1 ö  k 1
-
-
ç x k 1 , ÷ . Find 0 <  < s.t. B(x ,  )  G .
-
k
è 2 ø 2
k
k
k
 ¾¾¾® 0 and " k, B(x ,  k )  B(x ,  ).
k
k
k®¥
k–i
k–i
8.9 Review Questions
1. Discuss Completeness and Compactness.
2. Describe the Cantor's theorem.
3. Explain Baire category theorem.
4. Describe Compactness and Cantor set.

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