If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?

Question

If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?

If is not true in general, please give some counter-example "easy to check" (closed-form function).

Also, if possible, since could be relevant, I want to check the following two scenarios:

1. Unbounded domain $\mathbb{R}$: $\quad [a,\,b] \equiv (-\infty,\,\infty)$
2. Bounded domain $\in\mathbb{R}$: $\quad [a,\,b],\,\, -\infty<a<b<\infty$

I am trying to understand the consequences of having defined a function in a bounded domain, and for the absolute continuity definitions on Wiki I get a bit lost, since in general it is says is the condition required to made the Fundamental Theorem of Calculus to work (so I expect this to be true), but I don´t know if I missing something more abstract (since is also defined through measures), or If there exists some counterexamples that could defy this intuition.

Answer 1

If your function is absolutely continuous, it is continuous in particular. Thus it is absolutely integrable if the domain is bounded. If the domain is unbounded, that is not necessarily the case. Just take a non-zero constant on $[0,\infty)$.