Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain and let $v\in H^1(\Omega)$. Furthermore, let $S=(0,T)$ denote a time interval and let $s\in C^1(\overline{S\times\Omega},\overline{\Omega})$. For a smooth hypersurface $\Gamma\subset\Omega$, I want to investigate the continuity of a function $f\colon S\to\mathbb{R}$ defined via
$$f(t):=\int_\Gamma v(s(t,x))\,\mathrm{d}x.$$
I strongly suspect that this function is, indeed, continuous but, until now, failed to come up with a rigorous justification for this suspicion. Maybe a hint would already do the trick for me.
I came up with a solution to my own question: choosing a sequence of continuous functions $g_n$ approximating $v$ in $H^1(\Omega)$ (which is possible because of density). Then, estimating
$$|f(t_1)-f(t_2)|\leq\int_{\Gamma}\left|v(s(t_1,x))\pm g_n(s(t_1,x))\pm g_n(s(t_2,x))-v(s(t_2,x))\right|\,\mathrm{d}x\\ \leq\int_{\Gamma}\left|v(s(t_1,x))-g(s(t_1,x))\right|+\left|g_n(s(t_1,x))- g_n(s(t_2,x))\right|+\left|g_n(s(t_2,x))-v(s(t_2,x))\right| $$
and using the continuity of the $g_n$ and of $s$ yields the continuity.