Definition talk:Absolute Continuity

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I prefer to name the concept being defined as soon as possible, as otherwise one has to wait to see to which object the definition will apply. Also, writing "let..." for all objects appearing in the definition does not make any distinction between the previous things you need in order for the concept to make sense (the interval) and the actual object which has the property. Maybe it is a matter of style, but certainly it is the most standard in books.

Also, separating statments like "the following property holds" and then again "the following property holds" might be more confusing if it is not clear where "the following property" ends. I'd like to put some kind of box, or indent, the text with the property, but I don't know how to do it.--Cañizo 12:18, 19 February 2009 (UTC)

Okay, do it the way you want. Just that I like my mathematical proofs and definitions to take a breath every now and then. ;-)

You could define the bit about the disjoint closed real intervals separately, like:

"Given $\epsilon > 0$, let $S_\epsilon$ be a finite set of disjoint closed real intervals $\left[{a_1, b_1}\right], \ldots, \left[{a_n, b_n}\right] \subseteq I$ such that $\displaystyle \sum_{i=1}^n \left \vert {b_i - a_i} \right \vert < \epsilon$."

Then $f$ can be defined as:

$\displaystyle \forall \epsilon: \exists \delta: \forall S_\epsilon: \sum_{i=1}^n \left \vert {f \left({b_i}\right) - f \left({a_i}\right)} \right \vert < \delta$

or something, needs tidying, it's loose, but you get what I mean. --prime mover (talk) 21:28, 19 February 2009 (UTC)