Let $f$ be a continuous function such that $f(f(x))=x$

Question

Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that $f(f(x))=x$. I have to prove or disprove whether $f$ is identity or not.

Given conditions imply that $f$ is both injective and bijective. We know that a injective continuous function is monotone.

Assume on the contrary that $f$ is not identity then there exist a point $y$ such that $f(x)=y$ and $x \neq y$

Since $x$ is not equal to $y$, by order properties of reals there are two choices left. Either $x>y$ or $x<y$.

WLOG: Let us say $x>y$.

Suppose $f$ is monotonically increasing then $f(x)>f(y)$ which implies $y>x$ a contradiction. Suppose $f$ is monotonically decreasing then $f(x)<f(y)$ which implies $y<x$.

As I am not able to arrive at the contradiction, This lead me to believe that there are function which are continuous and satisfy the above condition but are not identity.

Am I right? Please give some examples.

Edits:

What additional condition can be put to ensure that $f$ is identity?

Is $f$ from $\mathbb R$ to itself? If it's from $\mathbb R^+\to\mathbb R^+$ then $f(x)=1/x$ would work. — YiFan Tey, Oct 28 '18 at 13:11
@user496634 I have edited the post. Function is from set of real numbers to set of real numbers. — Shweta Aggrawal, Oct 28 '18 at 13:16
BTW this question list quite a few examples: Functions that are their own inversion. (and other posts linked there might be of interest, too) A function which fulfills this condition is called involution. — Martin Sleziak, Oct 30 '18 at 07:53

score 9 · Accepted Answer · edited Oct 30 '18 at 07:50

9

For any continuous and invertible function $g(x)$,

$$f(x):=g^{-1}(-g(x))$$ fulfills the conditions.

E.g.

$$g(x):=x^3+1\to f(x)=-\sqrt[3]{x^3+2}.$$

edited Oct 30 '18 at 07:50

Martin Sleziak

53,687

answered Oct 28 '18 at 13:23

score 7 · Answer 2 · answered Oct 28 '18 at 13:12

7

You cannot get a contradiction because if $f(x)=-x$, then $f\bigl(f(x)\bigr)=x$.

answered Oct 28 '18 at 13:12

José Carlos Santos

427,504

score 2 · Answer 3 · answered Oct 28 '18 at 13:15

2

$$f(x)=-x\implies f(f(x))=f(-x)=-(-x)=x.$$

answered Oct 28 '18 at 13:15

YiFan Tey

17,431
4
28
66

Let $f$ be a continuous function such that $f(f(x))=x$

3 Answers3