Composition of Riemann integrable and continuous function is integrable

Question

Prove that if $f : [a,b] \to [c,d]$ is Riemann integrable , and $g: [c,d] \to \mathbb{R}$ is continuous then $g\circ f$ is integrable.

By Lebesgue we know because $f$ is integrable then $f$ must be discontinuous on at most a set of measure zero, so I need to show that $g\circ f$ is continuous except for at most a set of discontinuous points of measure zero.

I need some hints on how to do that, please help.

Answer 1

Hint:

If $f$ is continuous at $x$ and $g$ is continuous at $f(x)$, then $g\circ f$ is continuous at $x$.

Answer 2

By Lebesgue $f$ is continuous away from a set of measure zero . So there exists a subset $A$ of $[a , b]$ with measure $0$ such that the restriction of $f$ to $[a , b]\setminus A$ is continuous . So the composition map is continuous on $[a , b]\setminus A$ . Hence again by Lebesgue one can conclude that $ g \circ f $ is Riemann Integrable on $[a ,b]$ .