Functions of Bounded Variation

[jdcasey9]

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 \int_a^blf'(t)l dt. Hence, V^b_a\leq\int_a^blf'(t)ldt.

Homework Equations

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

The Attempt at a Solution

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

V₁(f,P) = \sumf(t_i+1)-f(t_i) = lf(a₁) - f(a)l

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

.
.
.

V_n(f,P) = \sumf(t_i+1)-f(t_i) = lf(b)- f(a_n-1)l

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

v₁(f,P)= lf(a)-f(a₁)l=\int_a^blf'(t)ldt = lf(a₁) -f(a)l

etc.

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

 

[jdcasey9]

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