Functions of Bounded Variation

In summary, we need to show that if a function f has a continuous derivative on a closed interval [a,b], and P is any partition of [a,b], then the variation V(f,P) of f on P is less than or equal to the integral of the absolute value of the derivative of f on [a,b]. This can be proven by dividing the function into monotone segments and treating each segment independently, ultimately resulting in V(f,P) being equal to the integral of the absolute value of the derivative of f on [a,b].
  • #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.
 
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  • #2
I can't get the cartesian product off of there, so please just ignore it.
 

What is meant by "bounded variation" in terms of functions?

Bounded variation refers to the behavior of a function in terms of how much it fluctuates or changes over a given interval. A function is considered to have bounded variation if the total change in its values over any interval is finite.

What is the significance of functions of bounded variation?

Functions of bounded variation are important in mathematical analysis and applications in fields like physics and engineering. They provide a measure of the smoothness of a function and can be used to describe the behavior of physical systems.

How is the total variation of a function calculated?

The total variation of a function f on an interval [a,b] is calculated by taking the supremum (or maximum) of the sum of the absolute values of the differences between consecutive function values at points in the interval.

What is the difference between functions of bounded variation and continuous functions?

While continuous functions are defined by their ability to have a continuous graph with no breaks or jumps, functions of bounded variation are defined by their smoothness and rate of change. A function can be continuous but not have bounded variation, and vice versa.

In what situations are functions of bounded variation useful?

Functions of bounded variation are useful in situations where we want to describe the behavior of a system or function in terms of its rate of change over a given interval. They are also important in the study of differential equations and Fourier analysis.

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