Linked Questions

10
votes
2answers
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!
14
votes
1answer
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$ ?
3
votes
2answers
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). ...
1
vote
1answer
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 $x\in\Bbb R$?
0
votes
2answers
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?
3
votes
1answer
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.
-2
votes
1answer
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 ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
12
votes
11answers
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 ...
7
votes
3answers
86 views

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 ...
1
vote
2answers
475 views

If $f(f(x)) = -x$ then is $f$ continuous?

let $f : \mathbb R \longrightarrow \mathbb R$ be a function such that $f (f(x)) = -x$ , $x \in \mathbb R$ then is $f$ continuous over $\mathbb R$? I have observed that $f$ is a bijection and so $f(x) ...
5
votes
2answers
224 views

How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$

How can I determine $ f(x)$ if $f(1-f(x))=x$ for all real $x$? I have already recognized one problem caused from this: it follows that $ f(f(x))=1-x $, which is discontinuous. So how can I construct ...
13
votes
1answer
484 views

Discontinuities of a function whose graph is invariant under rotation by 90 degrees

Prove that there is no function on open interval $(-1,1)$, which has only finite number of discontinuity point, such that its graph is invariant under rotation by the right angle around the origin.
4
votes
3answers
130 views

Find all functions $f\colon \Bbb R\to \Bbb R$ such that $f(1-f(x)) = x$ for all $x \in \Bbb R$ [duplicate]

Find all functions $f\colon \Bbb R\to \Bbb R$ such that $f(1-f(x)) = x$ for all $x \in \Bbb R$. This is a question from the national olympiad in Germany 2018. All i could do is to try with some ...

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