Functions of Bounded Variation

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The discussion focuses on proving that if a function f has a continuous derivative on the interval [a,b], then the variation V(f,P) for any partition P is less than or equal to the integral of the absolute value of the derivative over that interval. By dividing the function into monotone segments and calculating the variation for each segment, it is shown that the total variation can be expressed as an integral of the derivative. This leads to the conclusion that the total variation from a to b is bounded by the integral of the absolute value of the derivative. The proof effectively demonstrates the relationship between bounded variation and continuous derivatives. The discussion emphasizes the mathematical rigor required in establishing these properties.
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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)\leq \intablf'(t)l dt. Hence, Vba\leq\intablf'(t)ldt.

Homework Equations


Monotone function \subset BV[a,b]
\sumf(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) = \sumf(ti+1)-f(ti) = lf(a1) - f(a)l

V2(f,P) = \sumf(ti+1)-f(ti) = lf(a2) - f(a1)l

.
.
.

Vn(f,P) = \sumf(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=\intablf'(t)ldt = lf(a1) -f(a)l

etc.

Adding them all up V(f,P)= \intablf'(t)l dt, which satisfies our prompt.
 
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