in Spivak's proof of the inverse function theorem,
what is the definition of $\phi$?
in theorem 2.2:
$\phi$ is defined as $\phi(x)=f(x)-f(a)-\lambda (x-a)$. Is the definition of $\phi$ in the inverse function theorem the same as this? But then $a$ is not defined. (Spivak does define $a$ in the earlier part of the theorem but that $a$ does not seem to have any connection here)
Best Answer
$\phi (x)$ is just defined by that equation: $\phi (x_1-x)=f(x_1)-f(x)-\mu(x_1-x)$ or $\phi (t)=f(x+t)-f(x)-t\mu$.