Continuous dependence of the fixed point of a uniform contraction
= Continuous dependence of the fixed point of a uniform contraction
Suppose $T_\lambda$ are contractions of one complete metric space with a common constant $k<1$, and $T_\lambda x$ depends continuously on $\lambda$ for each fixed $x$. Their fixed points $x_\lambda$ depend continuously on $\lambda$, because for a fixed parameter $\lambda_0$,
$$
d(x_\lambda,x_{\lambda_0})
\leq\frac{d(T_\lambda x_{\lambda_0},T_{\lambda_0}x_{\lambda_0})}{1-k}.
$$