Let $T: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ be a continuous, linear and bijective mapping and let $f(\vec{x})$ be a continuously differentiable function, such that $$ \exists C > 0, \forall \vec{x} \in {\mathbb{R}}^{n}, ||f(\vec{x})|| \leq C|| \vec{x} ||^{2}$$ show that $$g(\vec{x}) : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$$
with
$$ g(\vec{x}) = T\vec{x} + f(\vec{x})$$ has a continuously differentiable inverse mapping in a neighborhood of $\vec{0}$.
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I think I need to show that the Jacobian of g(x) is regular, but how can I show that exactly and how to go on from there?
The Jacobian of $g(x)$ =
$$\begin{pmatrix} T_{11} + \frac{\partial f_1}{\partial x_1} &\dots &T_{1n} + \frac{\partial f_1}{\partial x_n} \\ \vdots & \vdots & \vdots \\ T_{n1} + \frac{\partial f_n}{\partial x_1} &\dots &T_{nn} + \frac{\partial f_n}{\partial x_n} \end{pmatrix}$$
edit: From the definition of the function it follows that $f(0)=0$, so also $(0)=0$ so does this mean that all the partial derivatives vanish from the above written jacobian? and only the derivatives of T, but how to show that the jacobian does not vanish?