Let $f$ be a Riemann integrable function defined on$[a,b]$ and let $g$ be a differentiable function with continuous on $[c,d]$.If the range of $g$ is contained in $[a,b]$,Is $f \circ g$ Riemann integrable on$[c,d]$？
If $g^{'}$ is non-zero on$[c,d]$,then the answer is positive.Go a step further ,let us remove this condition (as mentioned above),whether it will get the same conclusion？
Let $f\colon [-1,1]\to \Bbb R$ be defined by $f(x)=-1$ if $x\le0$ and $f(x)=1$ if $x> 0$.
Now let $g\colon [0,1]\to \Bbb R_+ \cup \{0\}$ be a smooth function which vanishes on a cantor set $C$ with positive measure.
Since Cantor sets are nowhere dense $f\circ g$ can not be continuous on $C$, but Riemann integrable functions have to be continuous almost everywhere.