The question is about bounded variation. When I trying to use the hint to fix this problem, I have no idea that how can I try to find F(x)=g(x)-h(x) and let g(x) and h(x) are monumentally increasing function Exercise 4. Let fe BV([0, a]) for some a > 0. Define-| $(t)dt, r€ (0, a],F(r):0,r = 0.Show that FE BV ([0, a]).(Hint: Jordan decomposition)Themem 3.2.2 CJandan Decom posicuon) tet ofbe a real-valuad funceion def imad an[a b] Then fEBV (Q,4) if and anly fly offo = g-hco for all xE [a, b] wheremomoton.cally Increasing9 a and h) arefunc tinson [e b].

Since it is given that f∈BV([0,a]), then according to Jordan decomposition theorem mentioned there…

Answered: Exercise 4. Let f e BV ([0, a)) for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The question is about bounded variation. When I trying to use the hint to fix this problem, I have no idea that how can I try to find F(x)=g(x)-h(x) and let g(x) and h(x) are monumentally increasing function

Exercise 4. Let fe BV([0, a]) for some a > 0. Define
-| $(t)dt, r€ (0, a],
F(r):
0,
r = 0.
Show that FE BV ([0, a]).
(Hint: Jordan decomposition)
Themem 3.2.2 CJandan Decom posicuon) tet of
be a real-valuad funceion def imad an
[a b] Then fEBV (Q,4) if and anly f
ly of
fo = g-hco for all xE [a, b] where
momoton.cally Increasing
9 a and h) are
func tins
on [e b].

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