Exercise 1.8. Suppose that $\mu$ is a continuous parameter and that for each $\mu \in[0,1]$, we have $f(z, \mu)=O(g(z, \mu))$ as $z \rightarrow z_0$ from $D$. The above proof suggests that it might be true that if the integrals exist in the Riemann sense for all $z$ close enough to $z_0$, then $$ \int_0^1 f(z, \mu) d \mu=O\left(\int_0^1|g(z, \mu)| d \mu\right) \quad \text { as } z \rightarrow z_0 \text { from } D, $$ since the integrals can be approximated by Riemann sums. Under what additional hypotheses on $f(z, \mu)$ and $g(z, \mu)$ can the proof be adapted to the continuous case? Can you find a counterexample?
I am reading Applied Asymptotic Analysis in the series Graduate Studies in Mathematics Volume 75. I'm confused by ex 1.8. I don't think this proposition is true but I can't find a counterexample. However, I find that when $f(z,\mu),g(z,\mu)$ is continuous in $D\times[0,1]$ and $g(z,\mu)\neq0$,the statement is true. Any attempt to find a counterexample will be helpful!
Here's what the notation means:
Definition 1.2 (Big-oh near $\left.z_0\right)$. Let $f(z)$ and $g(z)$ be two complex-valued functions defined in some set $D$ of the complex plane whose closure contains a point $z_0$ (that is, $z_0$ is a limit point of $D$ ). Then we write $$ f(z)=O(g(z)) \quad \text { as } z \rightarrow z_0 \text { from } D $$ if there is a number $\delta>0$ such that $$ f(z)=O(g(z)), \quad z \in D \text { with } 0<\left|z-z_0\right|<\delta, $$
Miller, Peter D., Applied asymptotic analysis, Graduate Studies in Mathematics 75. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4078-9/hbk). xv, 467 p. (2006). ZBL1101.41031.
You can take $$f(z,\mu) = \begin{cases}\mu^{z-1}, &z>0 \text{ and } \mu >0,\\ 0, & z\le 0 \text{ or } \mu = 0,\end{cases} \quad g(z,\mu) = 1, \quad D = [-1,1],\quad z_0=0.$$ Then for each fixed $\mu\in(0,1]$, we have $|f(\cdot,\mu)|\le \frac1\mu=O(1)$ (i.e. $f$ is bounded) near $z_0=0$, and similarly for $\mu = 0$, but $$ \int_0^1 f(z,\mu) d\mu = \begin{cases}\frac1{z},&z>0,\\ 0, &z \le 0,\end{cases}$$ which is unbounded as $z\to 0$.