1
vote
0answers
57 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
-1
votes
1answer
71 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 $x\in\Bbb R$?
3
votes
1answer
138 views

Functional equations leading to sine and cosine

This question is a possibly harder version of: Find $g'(x)$ at $x=0$. Question. Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, ...
19
votes
2answers
578 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that ...
0
votes
1answer
35 views

functional equation, find $h$ continous on $\mathbf{R}$ such that $h(x) + h(2x) + h(4x) = {x^n}$

I am having trouble finding easily $h$ defined and continuous on $\mathbf{R}$ verifying for all $x$ in $\mathbf{R}$, $$ h(x) + h(2x) + h(4x) = {x^n} $$ where $n$ is a fixed natural number. I have a ...
14
votes
1answer
331 views

Condition for an additive function to be continuous

The problem below is Problem 7 from this year's Miklos Schweitzer competition (contest ended Nov 4th). Suppose that $f: \Bbb{R} \to \Bbb{R}$ is an additive function (that is $f(x+y) = f(x)+f(y)$ ...
3
votes
1answer
98 views

Proving that a function is differentiable and equal to a constant value for all x

Let $f(x)$ denote a strictly positive continuous function defined on all real numbers with the property that $f(2012)=2012$ and $f(x)=f(x+f(x))$ for all $x$. Prove that $f(x)=2012$ for all $x$. I am ...
0
votes
1answer
36 views

Finding the function [duplicate]

I would appreciate if somebody could help me with the following problem: Q: $f(x):$ conti-function and $f(2x)-f(x)=x^3$ find $f(x)=?$
14
votes
5answers
371 views

If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?

Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
3
votes
1answer
173 views

Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)

If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
11
votes
4answers
374 views

$f(16x)=16f(x) $ and $ f$ is continuous

$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$. Should it be $f(x)=ax$? How can I prove that?
5
votes
2answers
85 views

$f,g:[a,b]\to [a,b]$ are continuous, $f\circ g=g\circ f$ and $f$ is 1-1. Show that $\exists t\in [a,b]$ so that $f(t)=g(t)=t$

Using the Intermediate Value Theorem on $h(x)=f(x)-x$ and $r(x)=g(x)-x$ I can easily show that $\exists t_f,t_g\in[a,b]$ so that $f(t_f)=t_f$ and $g(t_g)=t_g$. It remains to show that $t_f=t_g$. Since ...