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Suppose $f\left(x,\alpha\right)$ is a parameterized function.

$f:\,D\times\left[0,1\right]\rightarrow D$ where $D$ is a convex subset of $\mathbb{R}^{n}$.

Suppose $x^{*}$ is a fixed point of $f\left(x,1\right)$ . That is, $f\left(x^{*},1\right)=x^{*}$.

Under what conditions is $x^{*}$ also the limit of a the sequence of fixed points of $f\left(x,\alpha\right)$ as $\alpha\rightarrow1$ ?

Is continuity of $f$ sufficient?

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One useful example: $f(x,\alpha) = \alpha x$ (where $0 \in D$). Every point of $D$ is a fixed point of $f(x,1)$, but $0$ is the only fixed point of $f(x,\alpha)$ for $\alpha \ne 1$.

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  • $\begingroup$ Thank you, this is a nice simple couterexample. $\endgroup$
    – CommonerG
    Feb 17, 2016 at 18:36

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