I have an integral of the form: $$g(s)=\int_{D(s)}f(x)dx,$$ where $x\in\mathbb{R}^n$, $s\in\mathbb{R}$, $D(s)$ is a parameter-dependant subset of $\mathbb{R}^n$, $f$ is a "nice" function (i.e. $f\in\mathcal{C}^{\infty}$) and the integral converges for all $s$.
I want to understand if $g(s)$ is continuous, in particular in the case where $D(s)$ is a compact and convex set with non-zero measure for all $s$. All results I was able to find on the topic (e.g. https://encyclopediaofmath.org/wiki/Parameter-dependent_integral) consider a parameter-dependent integrand, with a constant domain of integration. I tried to transform the problem with: $$\int_{D(s)}f(x)dx = \int_{\mathbb{R}^n}f(x)\chi_{D(s)}(x)dx=\int_{\mathbb{R}^n}h(x,s)dx.$$ However, $h(x,s)$ is not continuous in $s$, meaning that I cannot apply any of the results I found.
If it can help, I am familiar with the concept of hemicontinuity of set-valued functions, which intuitively might be related to this issue (maybe hemicontinuity of $D(s)$ implies continuity of $g(s)$?).
Anybody knows any related results? Thanks!