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Functions of Bounded Variation

  • Thread starter jdcasey9
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  • #1
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1. Homework Statement [/b]
If f has a continuous derivative on [a,b], and if P is any partition of [a,b], show that V(f,P)[tex]\leq[/tex] [tex]\int[/tex]ablf'(t)l dt. Hence, Vba[tex]\leq[/tex][tex]\int[/tex]ablf'(t)ldt.

Homework Equations


Monotone function [tex]\subset[/tex] BV[a,b]
[tex]\sum[/tex]f(ti+1)-f(ti) = lf(b) - f(a)l



The Attempt at a Solution


Let P = {a=t0 < t1 < ... < tn}. So if we divide our function into monotone segments we have:

V1(f,P) = [tex]\sum[/tex]f(ti+1)-f(ti) = lf(a1) - f(a)l

V2(f,P) = [tex]\sum[/tex]f(ti+1)-f(ti) = lf(a2) - f(a1)l

.
.
.

Vn(f,P) = [tex]\sum[/tex]f(ti+1)-f(ti) = lf(b)- f(an-1)l

Then, treating this segments independently of the whole, we see that

v1(f,P)= lf(a)-f(a1)l=[tex]\int[/tex]ablf'(t)ldt = lf(a1) -f(a)l

etc.

Adding them all up V(f,P)= [tex]\int[/tex]ablf'(t)l dt, which satisfies our prompt.
 
Last edited:

Answers and Replies

  • #2
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I can't get the cartesian product off of there, so please just ignore it.
 

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