# Functions of Bounded Variation

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)$$\leq$$ $$\int$$ablf'(t)l dt. Hence, Vba$$\leq$$$$\int$$ablf'(t)ldt.

## Homework Equations

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

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

.
.
.

Vn(f,P) = $$\sum$$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=$$\int$$ablf'(t)ldt = lf(a1) -f(a)l

etc.

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

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