The proof i hopefully would be able to do. I have Absolutely no idea What kind of function would be non-integrable.

The fact that it's bounded and continuous almost seems to guarantee the functions integrability, the only thing i see destroying it is the open interval, however looking at it in the sense if Darboux Upper/Lower Sums, Sup{f(x)} and Inf{f(x)} need not belong to the interval, so even if the function achieves a max/min at the endpoints and not within the interval we can still integrate... maybe I'm thinking too hard.. i dunno. help!