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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_06” 2011/5/22 23:14 page 28 #28
6-28 Real Analysis
5. Suppose f
is continuous on [a, b] and >0 is given. Prove that there
exists a δ>0 such that
f (t) f (x)
t x
f
(x)
<
whenever 0 < |t x| , x, t ∈[a, b]. What about the converse?
Solution: Using the Mean Value Theorem, we have
f (t) f (x)
t x
=|f
(y )|
for some y between x and t. However by uniform continuity of f
(note
that continuous functions on compact sets are uniformly continuous),
given >0 there exists δ>0 such that |f
(s) f
(t)| <whenever
|s t| . Since
f (t) f (x)
t x
f
(x)
=|f
(y ) f
(x)| with |y x| < |t x| ,
the result ...
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Publisher Resources

ISBN: 9781299447561Publisher Website