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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_07” 2011/5/22 23:09 page 11 #11
Functions of Bounded Variation 7-11
Thus V
1
0
f 2a/(1 a), which implies that V
1
0
f = 2a/(1 a).
Hence f is of bounded variation on [0, 1]. However given any
subinterval of (0, 1), say [c, d] with c < d we can always find
points r of the form r
k
and points s, t which are not of the form
r
k
such that c < t < r < s < d. Then f (r)>0, f (s) = 0 and
f (t) = 0 and so f (t)<f (r) and f (s)<f (r). This shows that f
is neither decreasing nor increasing in the interval [c, d].
2. Here is an example of a function that is of bounded variation on
[,1] for each >0 but is not of bounded variation on [0, 1].
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Publisher Resources

ISBN: 9781299447561Publisher Website