In classical real analysis I've only seen absolute continuity defined for functions on compact interval $[a,b]$, where the two equivalent definitions are: $f:[a,b]\rightarrow\mathbb{R}$ is AC if
(1) Given $\epsilon > 0$ there is a $\delta > 0$ such that $\sum_{i=1}^n |f(y_i)-f(x_i)|< \epsilon$ for every finite collection of nonoverlapping intervals $( (x_i,y_i) )_{i=1}^n $ each contained in $[a,b]$ with $\sum_{i=1}^n |y_i-x_i|< \delta$. Or,
(2) $f'$ exists a.e. on $[a,b]$, $f'$ is integrable on $[a,b]$, and
$f(x) = \int_a^x f'(y) dy + f(a)$ for all $x \in [a,b]$.
Is there an accepted definition for absolute continuity of a function on an open, possibly unbounded, interval $(a,b)$ where $-\infty \leq a < b \leq \infty$?
It seems that definition (1) extends easily to this case if we replace $[a,b]$ by $(a,b)$. If we call this condition (1'), then it's easy to show (1') is equivalent to (see answer by Jonas below). A natural extension of (2) to this case would be
(2') $f$ is AC on an open set $U$ if, for all compact intervals $[c,d] \subset (a,b)$, $f$ is AC in the sense of (2) on $[c,d]$.
Which of these is the best extension of the definition? I don't know enough about the notion of absolute continuity of measures to know if my extended definitions are consistent with that generalization as well.