Absolutely continuous function on R

Question

What is the definition of absolute continuity in whole $\mathbb{R}$. I know the definition on an interval $[a, b]$. I have a trouble with understanding the definition of absolute continuity in whole $\mathbb{R}$.

Answer 1

Let I be any finite interval of $\mathbb{R}$. A function $f\colon I\to \mathbb {R}$ is absolutely continuous on I if for every positive number $\varepsilon$ , there is a positive number $\delta$ such that whenever a finite sequence of pairwise disjoint sub-intervals $(x_{k},x_{k+1})$ of I satisfies $$\sum _{k=1}^{n}\left(x_{k+1}-x_{k}\right)<\delta \mbox{ then}\displaystyle \sum _{k=1}^{n}|f(x_{k+1})-f(x_{k})|<\epsilon$$ in particular just take $I=[a_{n},b_{n}]\rightarrow \mathbb{R}$. Then you can think as restrictions $f_{n}:[a_{n},b_{n}]\rightarrow\mathbb{R}, $ with $f_{n}(x)=f(x) $ if for any $n\in\mathbb{N}$ $f_{n}$ is absolutely continuous we say that $f$ is continuous.

In words it means that when you restrict $f$ to any finite interval $[a,b]$ the restriction $f_{[a,b]}$ is absolutely continuous.