Skip to Main Content
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 page 13 #13
Limits and Continuity 5-13
by (f ± g)(x) = f (x) ± g(x), (fg)(x) = f (x)g(x) are all continuous
at x = a. In addition if g(x) = 0 for x S then the function f /g
defined by
f
g
(x) =
f (x)
g(x)
is also continuous at x = a.
Proof Follows easily by using Theorem 5.2.13 and the formulation of
continuity using limits.
Theorem 5.3.9 Let f : S T, g : T R be functions. Let a S, f
be continuous at a and g be continuous at f (a). Then g f is continuous
at a.
Proof Let x
n
a as n →∞. It is clear that f (x
n
) f (a) as n →∞
(using continuity of f at a) and g(f (x
n
)) g(f (a)) as n →∞(using
continuity
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Make: Calculus

Make: Calculus

Joan Horvath, Rich Cameron
Complex Analysis

Complex Analysis

ITL Education

Publisher Resources

ISBN: 9781299447561Publisher Website