643 views

### How to prove that $f(f(x))=-x$ implies that $f$ is not continuous? [duplicate]

I am trying to prove that: Given an $f:\mathbb{R} \rightarrow \mathbb{R}$, if $f(f(x))=-x$ then $f$ is not continuous? any help? Thank you!
539 views

### Is there a real-valued function $f$ such that $f(f(x)) = -x$? [duplicate]

Is there a function $f\colon \mathbb{R} \to\mathbb{R}$ such that $f(f(x)) = -x$ ?
357 views

### Do there exist functions such that $f(f(x)) = -x$? [duplicate]

I am wondering about this. A well-known class of functions are the "involutive functions" or "involutions", which have that $f(f(x)) = x$, or, equivalently, $f(x) = f^{-1}(x)$ (with $f$ bijective). ...
177 views

### Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$? [duplicate]

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?
105 views

### $f (f(x))=-x$ - there's no such function [duplicate]

Prove that there's no continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(f(x))=-x$. Any hints?
82 views

### Does there exist a continuous function $f :\mathbb{R} \rightarrow \mathbb{R}$ such that $f \circ f(x) = -x$ for all $x$? [duplicate]

Does there exist a continuous function $f :\mathbb{R} \rightarrow \mathbb{R}$ such that $f \circ f(x) = -x$ for all $x$? My feeling is that there isn't, but I don't know how to go about proving this.
93 views

### What can be $f$ so that $f(f(x)) = -x$? [duplicate]

What can be $f$ so that $f^2(x) = -x$ for all $x\in R$? I know that if $f^2(x) = -x$ then $f(x)$ is injective and $f$ can not be continuous. But I can not find an example of discontinuous function ...
69 views

### Function with the property $f\circ f=-id$ [duplicate]

Consider a functions $f:\mathbb R\to \mathbb R$ such that $f(f(x))=-x$ for all $x$. It is shown in f(f(x)) = − x, Windmills, and Beyond" by Martin Griffiths which appeared in Mathematics ...
36 views

### Determine all continuous real function which satisfies the following [duplicate]

We are required to determine all continuous real valued functions $f$ such that $$f(f(x))=-x$$ I’ve determined that if such a function exists, it must be bijective. But I don’t know if such a ...
2k views

### A function such that $f(f(n)) = -n$?

This question from somebody's job interview made me puzzled: Design a function f, such that: $f(f(n)) = -n$ , where n is a 32 bit signed integer; you can't use complex numbers arithmetic. If you ...
### Is it possible to define a function $f$ from positive real to positive real such that $f(f(x)) = {1 \over x}$
Is it possible to define a function $f$, from positive real to positive real such that $f(f(x)) = {1 \over x}$? The motivation comes from $1990$ IMO problem $4$, which one step involves defining such ...