Question

When is f: D ⊂ Rn → Rn called locally invertible? Is the function F given by
F(x+y+z , x − 2y + 3z, x + y −1) locally invertible at (0,1,2)?
Justify your answer.

Answer 1

Defenition: $f$ is locally invertible at $\vec{x}_0$ if there is a $\epsilon>0$ and a function $g\colon B_{\epsilon} (f (\vec{x}_0)) \to \mathbb{R}^n$ such that

$$f\circ g (\vec{y}) = \vec{y} \quad \forall \vec{y} \in B_{\epsilon} (f (\vec{x}_0))$$ $$g\circ f (\vec{x}) = \vec{x} \quad \forall \vec{x} \in B_{\epsilon} (\vec{x}_0)$$

Local invertibility refers to whether the closest affine approximation to $f$ at a given point $\vec{x}_0$ is invertible. The closest affine approximation to $f$ at a point $\vec{x}_0$ is given by $f(\vec{x}_0)$ plus the derivative of $f$ , otherwise known as the Jacobian in this case.

$$J(x,y,z)=\det \begin{pmatrix} 1 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 3 & 0 \end{pmatrix} =3$$

So, $F$ is locally invertible at any point.