Let $\mathbf{F} : (D \subset \mathbb R^N) \to \mathbb R^N$ be differentiable ($D$ is open), and let $\mathbf{J}$ be its Jacobian. Suppose that, for any $\mathbf x, \mathbf y \in D$, $\|\mathbf{J}(\mathbf x) - \mathbf J(\mathbf y)\|_2 \leq \gamma\|\mathbf x - \mathbf y\|_2$, $\gamma > 0$, and let $\mathbf J(\mathbf x)$ be nonsingular for all $\mathbf x \in D$, taking $\|\mathbf J(\mathbf x)^{-1}\|_2 \leq \beta$ for all $\mathbf x \in D$, $\beta > 0$. Consider the finite difference approximation to the $j$th column of the Jacobian, $$ \tilde{\mathbf J}_{*, j}(\mathbf x) = \frac{\mathbf F(\mathbf x + \epsilon \mathbf e_j) - \mathbf F(\mathbf x)}{\epsilon}, \quad j=1,2,\ldots,N, $$ where $\epsilon$ is sufficiently small, and $\mathbf e_j \in \mathbb R^N$ is the $j$th column of the $N$-square identity matrix $\mathbf{I}_N$.
I am interested in giving a bound on the inverse $\tilde{\mathbf J}(\mathbf x)^{-1}$ of the approximated Jacobian, $$ \tilde{\mathbf J}(\mathbf x) = \begin{bmatrix} \tilde{\mathbf J}_{*, 1} & \cdots & \tilde{\mathbf J}_{*, N} \end{bmatrix} = \frac{1}{\epsilon}\begin{bmatrix} \mathbf F(\mathbf x + \epsilon \mathbf e_1)-\mathbf F(\mathbf x) & \cdots & \mathbf F(\mathbf x + \epsilon \mathbf e_n)-\mathbf F(\mathbf x)\end{bmatrix}. $$ I have thought about using the fact that the error in the approximation in the $j$th column is given as $$ \|\mathbf J(\mathbf x)\mathbf e_j - \tilde{\mathbf J}(\mathbf x)\|_2 \leq \frac{\gamma}{2}\epsilon, \quad j=1,\ldots,N, $$ but am unsure how to actually apply this since I would like to identify if a bound $M$ does exist, and what it might be given by, for the inverse: $$ \boxed{\|\tilde{\mathbf J}(\mathbf x)^{-1}||_2 \overset{?}{\leq} M, \quad \forall \mathbf x \in D.} \tag{1}$$ Does anyone know any ways to construct such a bound, or any useful results for this?