Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}_n, \;\; \boldsymbol{x}(0) = \boldsymbol{\xi} \in \mathbb{R}^n
\end{equation}
where $\boldsymbol{f}(\cdot) \in \mathcal{C} \left(\mathcal{D} ; \mathbb{R}^n \right) $ and $\mathcal{D} \subseteq \mathbb{R}^n$ and $\mathcal{D}$ is open with $\boldsymbol{0}_n \in \mathcal{D}$, and $\boldsymbol{f}(\cdot)$ is locally Lipschitz apart from the origin $\boldsymbol{0}_n$. Let $\boldsymbol{\psi}\left(t, \boldsymbol{\xi} \right) $ be the unique solution of the above initial value problem. Suppose there exists $\mathsf{v}(\cdot) \in \mathcal{C}^{\infty}\left( \mathcal{D} ; \mathbb{R} \right) $ such that $\mathsf{v} (\boldsymbol{0}_n) = 0$ and
\begin{align}
& \forall \boldsymbol{\xi} \in \mathcal{D} \setminus \{ \boldsymbol{0}_n \} , \; \mathsf{v} (\boldsymbol{\xi}) > 0 \quad \text{and} \;\; \; \left. \dfrac{\partial\,\mathsf{v}\left(\boldsymbol{\psi}\left(t, \boldsymbol{\xi}\right) \right)}{\partial t} \right|_{t = 0}
= \dfrac{\partial \mathsf{v} (\boldsymbol{\xi})}{\partial \boldsymbol{\xi}} \boldsymbol{f}(\boldsymbol{\xi}) < 0,
\\[1mm]
& \forall \boldsymbol{\xi} \in \mathcal{D},\;\; \sigma (\boldsymbol{\xi}) : = \int_{\mathsf{v}(\boldsymbol{\xi})}^{0}
1 \left / \left. \dfrac{\partial\,\mathsf{v}\left(\boldsymbol{\psi}\left(t, \boldsymbol{\xi}\right) \right)}{\partial t} \right. \right|_{t = \vartheta(\tau, \boldsymbol{\xi})} \mathsf{d} \tau =
\int_{\mathsf{v}(\boldsymbol{\xi})}^{0}
\dfrac{1}{\left. \dfrac{\partial\,\mathsf{v}\left(\mathbf{x} \right)}{\partial \mathbf{x} }
\boldsymbol{f}\left( \mathbf{x} \right) \right|_{ \mathbf{x} = \boldsymbol{\psi} \left( \vartheta(\tau, \boldsymbol{\xi}), \boldsymbol{\xi} \right)} } \mathsf{d} \tau < + \infty
\end{align}
where $\vartheta(\cdot, \boldsymbol{\xi})$ with a given $\boldsymbol{\xi}$ is the inverse function of $[0, \sigma(\boldsymbol{\xi})) \ni t \mapsto \mathsf{v}\left( \boldsymbol{\psi}\left(t, \boldsymbol{\xi} \right) \right) $. Note that the inverse function of $[0, \sigma(\boldsymbol{\xi})) \ni t \mapsto \mathsf{v}\left( \boldsymbol{\psi}\left(t, \boldsymbol{\xi} \right) \right) $ is well defined since $\mathsf{v}\left( \boldsymbol{\psi}\left(t, \boldsymbol{\xi} \right) \right)$ is strictly decreasing and differentiable in $t$.
Question: Can we prove that $\sigma(\boldsymbol{\xi})$ is continuous at $\boldsymbol{\xi} = \boldsymbol{0}_n$?
Background: The aforementioned problem is encountered in the characterization of finite-time stability of differential equations. The function $\mathsf{v}(\cdot)$ is actually a Lyapunov function of the dynamical system, and $\sigma(\boldsymbol{\xi})$ is the setting-time of the system. ($\sigma(\boldsymbol{\xi})$ always has finite value which is why we say the system is finite-time stable)
For the scalar case $n = 1$, the continuity of $\sigma(\boldsymbol{\xi})$ at $\boldsymbol{0}_n$ can be proved via the application of Dominated convergence theorem without using $\mathsf{v}(\cdot)$. However, the multi-dimensional case is much more difficult to be deal with due to the presence of $\mathsf{v}(\cdot)$ and $\boldsymbol{\psi}(t, \boldsymbol{\xi})$.
Thank you so much!
