I have come across some problems of the form:
Given $X=(X_1,\dots,X_n)$ independent random variables with distribution in a parametric family $\{ F_\theta : \theta \in \Theta \}$, with density $f_\theta(x)$, defined in an interval dependent on $\theta$ (the expression for $f_\theta(x)$ is given), find a statistic that is sufficient and complete for $\theta$.
The standard procedure in these is to use Neyman-Fisher's factorization theorem to find a sufficient statistic, and prove it is also complete. So let $T$ be this statistic, and $f_\theta^T(t)$ its density.
Remember:
$T$ is complete if, for all $h: \mathbb{R}^k \to \mathbb{R}$ measurable, $$E_\theta[h\circ T \circ X] = 0 \quad \forall \theta\in\Theta \implies h = 0\quad P_\theta\text{-a.s.}\quad \forall \theta\in\Theta$$
Then, for $h$ measurable, this a typical case: $$E_\theta[h\circ T \circ X] = \int_{-\infty}^\theta h(t)f_\theta^T(t)dt = 0 \quad \forall \theta \in \mathbb{R}$$
Now, everywhere I've seen this solved, it is claimed that $h = 0$ by the FTC (differentiating both sides with respect to $\theta$), but $h$ need not be continuous, only measurable. My question is, is this argument correct? In affirmative case, why?
This step can be justified by this theorem.