Let $\|F'(\boldsymbol{\beta_1})-F'(\boldsymbol{\beta_2})\|_2 \leq M \| \boldsymbol{\beta_1}-\boldsymbol{\beta_2} \| _2,$ $\forall \boldsymbol{\beta_1},\boldsymbol{\beta_2} \in \overline{S(\boldsymbol{\beta}^*,\delta)},$ where \begin{equation}\overline{S(\boldsymbol{\beta}^*,\delta)}= \{ \boldsymbol{\beta} : \| \boldsymbol{\beta}^*-\boldsymbol{\beta} \|_2 \leq \delta \},\end{equation} and $\delta>0$. $\boldsymbol{\beta}^*$ is just the solution of the system $F(\boldsymbol{\beta})$ and $F'$ is its jacobian matrix. So $F'$ is Lipschitz. How can we prove that it holds:
$F(\boldsymbol{\beta})=F'(\boldsymbol{\beta})(\boldsymbol{\beta}-\boldsymbol{\beta}^*)+O(\|e\|_2^2)$ where $\|e\|=\boldsymbol{\beta}-\boldsymbol{\beta}^*$ ?