I am trying to show that given the vectors $\mathbf{a}$ and $\mathbf{b}$, the system of equations given by $$\mathbf{a}=\mathbf{x}_2 - \mathbf{x}_1$$ $$\mathbf{b}=\frac{\mathbf{x}_1}{\lVert \mathbf{x}_1 \rVert^3}-\frac{\mathbf{x}_2}{\lVert\mathbf{x}_2\rVert^3}$$ can be inverted to solve for $\mathbf{x}_1$ and $\mathbf{x}_2$ in a neighborhood around some point.
Assuming that $\mathbf{x}_1 \neq \mathbf{x}_2$, $\lVert \mathbf{x}_1 \rVert \neq \lVert \mathbf{x}_2 \rVert$, and $(\lVert \mathbf{x}_1 \rVert,\lVert \mathbf{x}_2 \rVert)$ are non-zero for the point of interest.
The problem with the Inverse Function Theorem is that it requires the functions to be continuously differentiable. Clearly, the second function is not continuously differentiable about $\mathbf{0}$.
If I were given the function $$\mathbf{c}=\frac{\mathbf{x}}{\lVert \mathbf{x} \rVert^3}$$ and asked to find $\mathbf{x}$ given $\mathbf{c}$ I know this can be inverted because it is easy to see that $\lVert \mathbf{x} \rVert = \lVert \mathbf{c} \rVert^{-1/2}$, but the Inverse Function Theorem would not be able to prove that this is invertible. Are there any other methods of determining whether or not a nonlinear function or system of nonlinear functions is invertible about a point?