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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_05” 2011/5/22 22:55 page8—#8
5-8 Real Analysis
are two possibilities: either f (x) = α or f (x) = α.Iff (x) = α, then
(g f )(x) = g(α) = β W .Iff (x) = α then f (x) T (V \{α})
and again g(f (x)) W . Thus for x S (U
1
\{a}), (g f )(x) W .
This shows that lim
xa
(g f )(x) = β.
For the converse we shall assume that lim
xa
(g f )(x) = β and
observe that if α =±∞then trivially (i) holds and hence there is
nothing to prove. Hence we are free to assume that α
R. In this
case, if (i) does not hold then given any neighbourhood U of a we
can find at least one point x S (U \{a}) with f (x) = α.By
taking U
n
= (a
1
n
, a
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Publisher Resources

ISBN: 9781299447561Publisher Website