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 ?