# Darboux integrable periodic function is integrable on any closed and bounded interval

How do I show that a Darboux integrable periodic function, with period $T$, is integrable on any closed and bounded interval?

Intuitively it is so clear to me, but how one can provide a proof?

Thank you for any help!

• Are you aware of the following standard fact? For $a < c < b$, a function $f: [a,b] \rightarrow \mathbb{R}$ is Darboux integrable on $[a,b]$ iff it is Darboux integrable on both $[a,c]$ and on $[c,b]$. If so there's not much left to show. – Pete L. Clark Apr 27 '13 at 3:17