Here an exercise that I need to work on, and I have some difficulties:
" Let g and f be defined on closed intervals and the range of f is contained in the domain of g so that g o f is defined. (1) Show that if f and g are integrable does not imply that g o f is integrable. (2) If f is increasing and g integrable, is g o f integrable? (proof or counterexample) (3) If f is integrable and g increasing, is g o f integrable? (proof or counterexample)"
For (1), I found a counterexample with f(x)=0 if x=0 or =1 otherwise and g(x)=0 if x is not rational or =p/q if w is rational. Both f and g are integrable (but I don't know how to prove it) but the composite is not.
My intuition is that (2) is false and (3) is true, but I have no idea what the proof and counterexample are.
Cours you help me finishing that please?