Prove Lipschitz function composed with an integrable function is integrable on [a, b]

Question

Given a Lipschitz function $g$ (i.e. $|g(x) - g(y)| \leq L |x - y|, \forall x, y \in dom(g)$), and an function $f$ integrable on $[a, b]$, how do we prove $g \circ f$ is integrable on $[a, b]$, preferably using Darboux integrals/sums?

Let us assume $g, f$ have appropriate domains and codomains.

Answer 1

Note that if $M=\sup_{x \in [a,b]} f(x), m = \inf_{x \in [a,b] } f(x)$, then \begin{eqnarray} \sup_{x \in [a,b]} g(x) -\inf_{x \in [a,b] } g(x) &=& \sup_{x,y\in [a,b]} g(x)-g(y) \\ &=& \sup_{x,y\in [a,b]} |g(x)-g(y)| \\ &\le & L \sup_{x,y\in [a,b]} |f(x)-f(y)| \\ &\le & L \sup_{x,y\in [a,b]} f(x)-f(y) \\ &=& L (M-m) \end{eqnarray}

Now consider a partition $P$ such that $U(f,P) - L(f,P) < { \epsilon \over L}$ and use the above to estimate $U(g\circ f,P) - L(g \circ f,P)$.

Answer 2

Hint: A function is (Riemann) integrable if and only if its set of discontinuity has measure zero.