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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 15 #15
Differentiation 6-15
Since x > 0 is arbitrary we have f (x) = cos x 1 +
x
2
2
> 0or
that cos x > 1
x
2
2
.
Theorem 6.3.15 If f is differentiable on [a, b], then all the disconti-
nuities of f
are of second kind.
Proof We shall show that if f
(y+) exists for some y ∈[a, b], then
f
(y+) = f
(y) and similarly if f
(y) exists for some y ∈[a, b], then
f
(y) = f
(y) (this will show that there are no discontinuities of first
kind for f
in [a, b]). By definition
f
(y) = lim
h0
f (y + h) f (y)
h
.
We are free to allow h 0 with the condition that h > 0. Applying
the Mean Value Theorem we have
f
(y) = lim
h0,h>
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Publisher Resources

ISBN: 9781299447561Publisher Website