Let $X$ a compact set and $A\subseteq \mathbb{R}$. Consider a continuous function $f\colon X\times A\to X$ and construct a fixed-point iteration as follows $$ x_{k+1}=f(x_k,a),\quad x_0\in X, a \in A.\quad (\star) $$

My question: If $(\star)$ admits a unique fixed point, denoted by $\mathrm{Fix}(f_a)$, for all $a\in A$, can we conclude that $\mathrm{Fix}(f_a)$ is a continuous function of $a$? What can be said in case $(\star)$ admits a set of fixed points for all $a\in A$?

Thanks for your help.

Comments. This question is different from this one. Indeed, here $f$ is not assumed to be contractive.


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