Riemann Integrabe Step Functions

[Arkuski]

Suppose that g:[a,b]\rightarrow[c,d] is Riemann integrable on [a,b] and f:[c,d]\rightarrow ℝ is Riemann integrable on [c,d]. Prove that f\circ g is Riemann integrable on [a,b] if either f or g is a step function.

The proof for g being a step function seems easy enough, but the other way seems much trickier. Thoughts?

 

[Bacle2]

Have you tried doing a formal composition:

c_iχ_Ai °g

And then extending by linearity to

Ʃ_k=1ⁿc_kχ_Akog ?