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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]

Monotone function [tex]\subset[/tex] BV[a,b]

[tex]\sum[/tex]f(t

Let P = {a=t

V

V

.

.

.

V

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

v

etc.

Adding them all up V(f,P)= [tex]\int[/tex]

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]

_{a}^{b}lf'(t)l dt. Hence, V^{b}_{a}[tex]\leq[/tex][tex]\int[/tex]_{a}^{b}lf'(t)ldt.## Homework Equations

Monotone function [tex]\subset[/tex] BV[a,b]

[tex]\sum[/tex]f(t

_{i+1})-f(t_{i}) = lf(b) - f(a)l## The Attempt at a Solution

Let P = {a=t

_{0}< t_{1}< ... < t_{n}}. So if we divide our function into monotone segments we have:V

_{1}(f,P) = [tex]\sum[/tex]f(t_{i+1})-f(t_{i}) = lf(a_{1}) - f(a)lV

_{2}(f,P) = [tex]\sum[/tex]f(t_{i+1})-f(t_{i}) = lf(a_{2}) - f(a_{1})l.

.

.

V

_{n}(f,P) = [tex]\sum[/tex]f(t_{i+1})-f(t_{i}) = lf(b)- f(a_{n-1})lThen, treating this segments independently of the whole, we see that

v

_{1}(f,P)= lf(a)-f(a_{1})l=[tex]\int[/tex]_{a}^{b}lf'(t)ldt = lf(a_{1}) -f(a)letc.

Adding them all up V(f,P)= [tex]\int[/tex]

_{a}^{b}lf'(t)l dt, which satisfies our prompt.
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