Fixed Points are Closed

Question

Prove that the set of fixed points of a continuous function f defined on R is closed.

This is specific to the Euclidean Space.

score 8 · Answer 1 · answered Oct 24 '15 at 18:22

8

If $f$ is continuous, so is $g$ given by $g(x)=f(x)-x$.

The set of fixed points of $f$ is the same as the set of zeros of $g$, and so is closed because $g$ is continuous.

answered Oct 24 '15 at 18:22

lhf

See also https://math.stackexchange.com/questions/55846/is-the-set-of-fixed-points-in-a-non-hausdorff-space-always-closed – lhf Oct 16 '22 at 19:03

score 2 · Accepted Answer · answered Oct 24 '15 at 18:13

2

Suppose $x_{n}$ are fixed points for $f$, and $x_{n} \to x$. What can you say about $f(x)$?

Hint:

$$f(x) = f(\lim_{n} x_{n}) = ... ?$$

answered Oct 24 '15 at 18:13

Elchanan Solomon

2 Answers2