Riemann integrability of composite functions

[darkreignmoon]

Hi, I'm stuck on this problem here about composite function, help is appreciated:

Let g : [a,b] -> [c,d] be Riemann integrable on [c,d] and f : [c,d] -> R is Riemann integrable on [c,d]. Prove that f o g is Riemann integrable on [a,b] if either f or g is a step function

I was able to solve the part where g is a step function but if f is a step function then I don't know how to approach this.

 

[Eynstone]

Consider the set of discontinuities of fog. f(g(x)) can be discontinuous when either g is discontinuous(say on the set G) or when it attains a 'bad' value in the domain of f(say in F).
Both F,G have measure 0 as f,g are Riemann integrable.FXG has measure 0 if f is a step function & hence the result.
I wonder what happens in case of Lebesgue integration.