TITLE: Continuity of Fixed Point
QUESTION [5 upvotes]: For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq (1-\epsilon) \left\| x-y\right\|$ for all $x,y \in \mathbb{R}^n$.
Assume that for all $x \in \mathbb{R}^n$, the mapping $a \mapsto f_a(x)$ is continuous.
Now let $x_0 \in \mathbb{R}^n$ be fixed. 
For all $a \in \mathbb{R}$, define the sequence $\{ x_a^0, x_a^1, x_a^2, \cdots \} := \{ x_0, \ f_a(x_0), \ f_a( f_a(x_0) ), \ \cdots \}$, that is, $x_a^k := f_a^{(k)}(x_0)$ for all $k \geq 0$. Define $ \bar{x}_a := \lim_{k \rightarrow \infty} x_a^k$.
Under what additional conditions the mapping $a \mapsto \bar{x}_a$ is continuous?

REPLY [7 votes]: Under the assumption that $\epsilon$ is constant across $a$, then no additional assumptions are needed. 
Suppose that $x_a$ is the fixed point of $f_a$, and choose $e>0$.  Then there is a $\delta$ such that for all $b$ within $\delta$ of $b$, you have $e\epsilon\geq \vert f_b(x_a)-f_a(x_a)\vert=\vert f_b(x_a)-x_a\vert$. 
Now, repeatedly apply $f_b$ to $x_a$.  If $x_b$ is the fixed point of $f_b$, then 
$$\vert x_b-x_a\vert\leq(e\epsilon)\sum_{i=0}^{\infty}(1-\epsilon)^{i}=\frac{e\epsilon}{\epsilon}=e.   $$