What is the derivative of total variation?

Let f(x) be a function of bounded variation on closed, bounded interval [a, b].

Let g(x) = V(f, [a, x])

where given some partition of [a, x].

How can I calculate the derivate of g(x)?

I know of differentiation term-by-term but not sure if this would apply here.

Thanks

Re: What is the derivative of total variation?

I would start with the definition of the derivative and see what you get:

$\displaystyle g'(x) = \lim_{h \to 0} \dfrac{g(x+h) - g(x)}{h}$

Re: What is the derivative of total variation?

This is what I get

If I consider the supremum of g' over all partitions of [a, b], is it true that ?

For example, if my partition is just two points then I get

Re: What is the derivative of total variation?

No, quite the contrary. Instead, you just showed that if $g'(x)$ exists, then $g'(x) \ge |f'(x)|$ since you want the supremum over all possible partitions (one of which is the partition with just two points).

Re: What is the derivative of total variation?

Yes, sorry, that's what I meant to write.

Thanks Slip.