I've been looking at how integrable functions behave under composition, and I know that if f and g are integrable, f(g(x)) is not necessarily integrable, but it -is- necessarily integrable if f is continuous, regardless of whether g is. So I was wondering, what about if g is continuous and f wasn't, rather than f was? Is there an example of a discontinuous integrable f and a continuous integrable g such that f(g(x)) is non-integrable?
I don't want an explicit proof of the answer, but I'd just like to know whether there is or isn't such an example, so I can begin looking for a counterexample or a proof as appropriate, rather than ending up trying to prove something which is false or look for a counterexample for something which is true. My intuition tells me there isn't a counterexample, but I've found relying on intuition in analysis is a very bad idea!