There is a problem with the OP.
There is a theorem (its proof is quite tedious): If is integrable on then is integrable on .
In the OP the finite interval was not included. So I am not sure what is meant.
However, in the example given by Bruno J is not integrable but is integrable. The inverse of the OP.
This is the definition I know:
is integrable if . Now in Bruno J's example I don't think either or are integrable since and are both identically 1 and the integral is not finite.
With regards to Plato's post, well it tells us that we can't use a function integrable on [a,b] since then is going to be integrable. But any function defined on a subset of should do. So still thinking.....