Construct a nonconstant increasing function $\alpha$ and nonzero continuous function $f\in R_\alpha[a,b]$ such that $\int_a^b|f|d\alpha=0$. Is it possible for $\alpha$ to be continuous？
Question from Carothers's Real Analysis. I have no I idea how to approach this. My gut feeling is that this is not possible since if $f$ is non-zero and continuous, $|f| > 0$. Thus, for any partition, we have $L(f, P) = \sum_i m_i\Delta\alpha_i$ where $m_i > 0$ and $\Delta\alpha_i > 0$ for at least some i. Thus, this may contradict with $\int_a^b|f|d\alpha=0$. However, I'm not sure how does continuous plays a role here.