It seem correct in gereral. (I have not carefully read it).
But I notice you may want to avoid divison by zero.
Suppose is Riemann integrabe on . Let be any number in , and let be any real number different from . Define a function as follows: . Prove that is Riemann integrable on and that .
Proof. Let . Choose . Let be a partition of with . Choose arbitrarily such that . Then .
So . Thus . Then . Hence is Riemann integrable on and .
Is this correct?
Let be integrable on then is integrable on and .
If you know this fact then you can use it to prove your problem.
Lemma: Let be a zero function on except at finitely many points, then is integrable with .
Proof: Let the non-zero points be so that and let . For and a partition form the Riemann sum where . Notice that . This can be made arbitrary small if is small enough. Thus, .
To prove you problem, consider and use the lemma above.