The famous Brower's fixed-point theorem states that any $ f $ function that maps a compact and convex set itself has a fixed point.
I would like to know if minor disturbances in the function $ f $ could only cause minor disturbances in the fixed point $ x $ of $ f $. In other words the question would be as follows. If a function $ g $ is close to $ f $ then will the fixed points of $ f $ be close to the fixed points of $ g $?
One problem with this question is that the number of fixed points of $ g $ may be greater or less than the number of fixed points of $ f $. Thus, there may be a fixed point $ x_f $ of $ f $ such that $ g (x) \neq x $ to $ x $ in some neighborhood of $ x_f $.
However the Brouwer's fixed-point theorem guarantees that the number of fixed points of $ g $ is always greater than or equal to $ 1 $. Thus, the question could be improved and put in the following terms. In the set $ \mathrm{Fix}(f) $ of fixed points of $ f $ there would be $ x_f \in \mathrm{Fix} (f) $ such that if any $ g $ application is close to $ f $ there would be $ x_g \in \mathrm{Fix }(g) $ tal which $ x_f $ is close to $ x_g $?
Technically the question would be as follows. Let $ \Omega \subset \mathbb{R}^n$ be a compact and convex set. Rig the set $ C^0(\Omega, \mathbb{R}^n) $ with the supreme norm $ \| f \|_{\infty}: = \sup \{| f (x) |: x \in \Omega \} $. Take $ f \in C^0(\Omega, \mathbb{R}^n)$ such that $ f (\Omega) \subseteq \Omega $. Given $ \epsilon> 0 $ and $ g $ satisfying the condition $ \| f-g \|_\infty <\epsilon $, with $ g(\Omega) \subseteq \Omega $, there is $ x_f \in \mathrm{Fix} (f) $, $ \delta> 0 $ and $ x_g \in \mathrm{Fix} (g) $ such that $ \| x_f-x_g \| <\delta $? How to prove it?