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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 page5—#5
Differentiation 6-5
Proof For t S \{x}, we have
f (t) = (t x)
f (t) f (x)
t x
+ f (x)
and hence lim
tx
f (t) = 0 · f
(x) + f (x) = f (x). This completes the
proof.
Note 6.2.6 The above theorem shows that every differentiable function
is continuous. However the converse is not true as seen by the following
example.
Example 6.2.7 f :
R R defined by f (x) =|x|is continuous (infact
it is even uniformly continuous as seen by the inequality ||x|−|y||
|x y| <if |x y| = ). But lim
x0
f (x)f (0)
x0
does not exist because
this limit is equal to 1 if x 0 with the condition x > 0 and is equal
to 1ifx
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Publisher Resources

ISBN: 9781299447561Publisher Website