Q:

Differentiability of fixed point

I am trying to review some past hw problems. I was never able to figure out how to do this problem. Can anyone help me out at all? Thanks.

A:

Fixed $\epsilon $, let be $f_\epsilon R \to R$ given  by $f_\epsilon(x) = f(x, \epsilon)$. Follows of hipotesis that is contraction and therefore exist $g(\epsilon) \in R$ such that  $f_\epsilon(g(\epsilon) ) = g(\epsilon)$.
We showing now that $g$ is differential :
$$g(\epsilon + h) - g(\epsilon) = f( g(\epsilon + h), \epsilon + h) - f( g(\epsilon ), \epsilon) = (\frac{\partial f}{\partial x}( g(\epsilon ) \epsilon) + \frac{\partial f}{\partial y}( g(\epsilon ))h + r((h,0))$$
where $\|r(h,0)\| \to 0$ when $h \to 0$, this implies that $g'(\epsilon) = \frac{\partial f}{\partial x}(g(\epsilon),\epsilon) + \frac{\partial f}{\partial y}(g(\epsilon),\epsilon)$.
Since, $g$ is differential, we have that $g$ is continuos and because composition of continuos functions it's continuos, we have that $\frac{\partial f}{\partial x}(g(\epsilon),\epsilon) + \frac{\partial f}{\partial y}(g(\epsilon),\epsilon)= g'(\epsilon)$ is continuos.

