I'm working on a problem, stated as follows:
If $f$ is integrable with respect to every bounded, monotone increasing function $g$ on $[a, b]$, then is $f$ continuous on $[a, b]$?
I have proved that the converse is true, I want to know whether this statement is correct. This is my work until now. Forgive me for my little work.
Since $g$ is monotone, increasing, Riemann's condition holds, so the term $$\sum_{j=1}^p (M_j-m_j)(g(x_j)-g(x_{j-1}))$$
could be made arbitrarily small for a partition $\pi=\{x_0, x_1,...x_p\} \in \Pi[a, b]$. The choice of $g$ is also arbitrary, so the difference $g(x_j)-g(x_{j-1})$ could also be made arbitrarily small. This would imply that the difference between $M_j$ and $m_j$ could be arbitrarily small. So I suspect that $f$ should be uniformly continuous. However, I don't know whether this logic is fine, and how to formally write this proof down. Thanks in advance.