As a function of (lambda), show that the transformation (mathbb{R}_{+}^{n} ightarrow mathbb{R}^{n}) [ (g y)_{i}=frac{y_{i}^{lambda}-1}{lambda dot{y}^{lambda-1}} ]

Question:

As a function of \(\lambda\), show that the transformation \(\mathbb{R}_{+}^{n} ightarrow \mathbb{R}^{n}\)

\[
(g y)_{i}=\frac{y_{i}^{\lambda}-1}{\lambda \dot{y}^{\lambda-1}}
\]

is continuous at \(\lambda=0\), and that the Jacobian is the absolute value of

\[
\frac{1}{\lambda}+\frac{\lambda-1}{n \lambda} \sum y_{i}^{-\lambda} \text {. }
\]

Find the limits for \(\lambda=0, \pm 1\). Discuss the implications regarding invertibility? For \(\tau>0\), show that \(g(\tau y)\) is not expressible in the form \(a+b g(y)\) for any constants \(a, b\) depending on \(\tau, \lambda\).