Let $g$ be a Riemann integrable function on $[a,b]$, and $f$ is a continuous. Prove that $f(g(x))$ is Riemann integrable for all $x\in[a,b]$. Let $g$ be a Riemann integrable function on $[a,b]$, and assume $f$ is a continuous function defined on $g(x)$, for all $x\in[a,b]$. Prove that $f(g(x))$ is Riemann integrable for all $x\in[a,b]$.
What I have so far: 
I know that if $f(g)$ is continuous then certainly Riemann, but this requires $g$ to be continuous. So consider the definition of Riemann integrable: for all $\epsilon>0$ there exists a partition $P$ such that the difference of upper sum and lower sum is bounded by this $\epsilon$. Since $g$ is Riemann integrable, there exists a partition such that $U(P,g)-L(P,g)<\epsilon$. Since $f$ is continuous, there exists $\delta>0$ such that the difference of upper sum and lower sum over any interval of length $\delta$ is bounded by $\epsilon$. I'm having trouble of combining these two together...
 A: EDIT: I wrote the solution below under the implicit assumption that $f$ is defined and continuous on $\mathbb{R}$. But this is not clearly stated in the question, and if absolute continuity does not hold the boundness need not hold either, as rightfully pointed out by @KilluaZoldyck

Use Lebesgue criterion for Riemann integrability:
$g$ is Riemann integrable iff $g$ is bounded and the sets of its discontinuities has Lebesgue measure zero.
$f$ is continuous, so $f\circ g$ is bounded. Moreover $\operatorname{disc}(f\circ g)\subseteq \operatorname{disc}(g)$, so the set of discontinuities of $f\circ g$ has Lebesgue measure zero.
So $f\circ g$ is Riemann integrable.
A: Consider a partition $a = x_0 < x_1 < \ldots < x_n = b$ and for a subinterval $I_j = [x_{j-1},x_j]$ define 
$$D_{j}(g) = \sup_{x \in I_j}g(x) -  \inf_{x \in I_j}g(x) = \sup_{x,y \in I_j}|g(x) - g(y)| \\ D_{j}(f \circ g ) = \sup_{x \in I_j}f(g(x)) -  \inf_{x \in I_j}f(g(x)) = \sup_{x,y \in I_j}|f(g(x)) - f(g(y))|$$
Since $f$ is continuous it is uniformly continuous and bounded (by extension if necessary) on a closed interval $[c,d]$ such that $g([a,b]) \subset [c,d].$
Hence, $|f(x)| \leqslant M$ for $x \in [c,d]$ and for every $\epsilon >0$ there exists $\delta > 0$ such that if $|x_1 - x_2| < \delta$ then $|f(x_1) - f(x_2)| < \epsilon/(2(b-a))$ .
Since $g$ is integrable, if the partition norm $\|P\|$ is sufficiently small we have 
$$U(P,g) - L(P,g) = \sum_{j=1}^n D_j(g) (x_j - x_{j-1}) < \frac{ \delta \epsilon}{4M}.$$
We can split the upper-lower sum difference $U(P,f \circ g) - L(P, f \circ g)$ into two sums as given by
$$\tag{1}U(P,f \circ g) - L(P, f \circ g) = \sum_{D_j(g) \geqslant \delta} D_j(f \circ g)(x_j - x_{j-1}) + \sum_{D_j(g) < \delta} D_j(f \circ g)(x_j - x_{j-1})$$
In the second sum on the RHS of (1) we have $D_j(f \circ g) < \epsilon/(2(b-a)$ since by uniform continuity $D_j(g) < \delta \implies |g(x) - g(y)| < \delta  \implies |f(g(x)) - f(g(y))| < \epsilon/(2(b-a)$ for all $x,y \in I_j$.
Thus,
$$\tag{2}\sum_{D_j(g) < \delta} D_j(f \circ g)(x_j - x_{j-1}) < \frac{\epsilon}{2}$$
Considering the first sum on the RHS of (1), first note that
$$\sum_{D_j(g) \geqslant \delta} (x_j - x_{j-1}) \\ < \delta^{-1}\sum_{D_j(g) \geqslant \delta} D_j(g)(x_j - x_{j-1})  < \delta^{-1} [U(P,g) - L(P,g)]  < \delta^{-1} \frac{\delta \epsilon}{4M} = \frac{\epsilon}{4M} .$$
Hence,
$$\tag{3}\sum_{D_j(g) \geqslant \delta} D_j(f \circ g)(x_j - x_{j-1}) < \sum_{D_j(g) \geqslant \delta} 2M(x_j - x_{j-1}) < \frac{\epsilon}{2}.$$
From (1), (2) and (3) we obtain
$$U(P,f \circ g) - L(P, f \circ g) < \epsilon,$$
and conclude that $f \circ g$ is integrable.
