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?
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 $\alpha \ne 1$.