1. **Problem Statement:** Determine if the composition of two Riemann integrable functions is always Riemann integrable. 2. **Recall Definitions:** A function $f$ is Riemann integrable on an interval $[a,b]$ if the set of its discontinuities has measure zero. 3. **Key Fact:** If $f$ and $g$ are Riemann integrable on $[a,b]$, it does not necessarily imply that $f \circ g$ is Riemann integrable. 4. **Explanation:** The composition $f(g(x))$ may fail to be Riemann integrable if $g$ maps a set of positive measure to a set where $f$ is highly discontinuous. 5. **Example:** Consider $g$ as the identity function (which is integrable) and $f$ as the Dirichlet function (which is not integrable). Then $f \circ g = f$ is not integrable. 6. **Conclusion:** The composition of Riemann integrable functions is not always Riemann integrable. Hence, the answer is **No**, the composition of Riemann integrable functions is not always Riemann integrable.