Consider a continuous map $F:(a,b)\times\mathbb{S}^n\to\mathbb{R}^{n+1}$ such that for any $t\in(a,b)$, the map $F(t,\cdot)=F_t:\mathbb{S}^n\to\mathbb{R}^{n+1}$ is Lipschitz continuous. The $n$-Hausdorff measure of $F_t(\mathbb{S}^n)$ is finite due to the Lipschitz continuity of $F_t$. My question is, whether the volume function $$ \mathrm{Vol}(t) = \mathcal{H}^{n}(F_t(\mathbb{S}^n)) $$ is continuous?

Based on the experience of the length functional on continuous curves, my guess is that the volume function $\mathrm{Vol}(t)$ is lower semicontinuous. However, I also want to know under which condition can I get a continuous function $\mathrm{Vol}(t)$.

Consider a continuous map $F:(a,b)\times\mathbb{S}^n\to\mathbb{R}^{n+1}$ such that for any $t\in(a,b)$, the map $F(t,\cdot)=F_t:\mathbb{S}^n\to\mathbb{R}^{n+1}$ is Lipschitz continuous. The $n$-Hausdorff measure of $F_t(\mathbb{S}^n)$ is finite due to the Lipschitz continuity of $F_t$. My question is, whether the volume function $$ \operatorname{Vol}(t) = \mathcal{H}^n(F_t(\mathbb{S}^n)) $$ is continuous?

Based on the experience of the length functional on continuous curves, my guess is that the volume function $\operatorname{Vol}(t)$ is lower semicontinuous. However, I also want to know under which condition can I get a continuous function $\operatorname{Vol}(t)$.