# Prove variation of a differentiable function

1. Mar 20, 2013

### stripes

1. The problem statement, all variables and given/known data

Prove that if a function f is once-differentiable on the interval [a, b], then

$Vf = \int ^{b}_{a} | f'(x) | dx,$

where $Vf = sup_{P} \sum^{i=n-1}_{0} | f(x_{i+1}) - f(x_{i}) |$ where the supremum is taken over all partitions $P = \left\{ a = x_{0} < x_{1} < ... < x_{n} = b .\right\}$

Use Taylor's Theorem (or the mean value theorem) and the definition of the integral.

Hint, adding an extra point to a partition cannot decrease $\sum^{i=n-1}_{0} | f(x_{i+1}) - f(x_{i}) |$

In our class, we use this definition for a Riemann integral:

2. Relevant equations

3. The attempt at a solution

Well I've gotten as far as writing down the definition of the integral and the definition of the total variation. Not sure how to apply the mean value theorem here. I did apply it on a later question but we were given a function to work with. Not sure how to start this one.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution