In Pugh's Real Mathematical Analysis (first edition), ch. 3, exercise 41 (b) (p. 195):
Assume that $\psi: [c, d] \to [a, b]$ is continuously differentiable. Prove that if $f \circ \psi$ is Riemann integrable for each Riemann integrable $f$ on $[a, b]$, then the critical points of $\psi$ form a zero set.
But if $\psi$ is constant, then it is everywhere critical and thus its critical points are certainly not a zero set (assuming $c < d$ of course). Yet $f \circ \psi$ is then constant for all $f$, and thus Riemann integrable. So this seems to be a counterexample. Have I done something wrong?