The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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15
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2answers
1k views

Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?

If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
2
votes
2answers
75 views

Finding Value, Related To Functional Equation

$f(x)$ is continuous for $\forall x \in R$ and $f(2x)-f(x)=x^{3}$ (1) $f(x)+f(-x)$ is constant ? (2) $f(0)=0$ ? I don't know how to use the continuity. especially for $f(0)=0$ ?
5
votes
2answers
528 views

$f: \Bbb N→ \Bbb N$ , $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$

How to find all functions $f: \Bbb N→ \Bbb N$ which satisfy $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$ ($\Bbb N$ is the set of all natural numbers, i.e. positive integers) ?
1
vote
1answer
82 views

Cauchy's Problem

I am looking at the Cauchy's functional equation here: http://en.wikipedia.org/wiki/Cauchy's_functional_equation. Could someone help me on how to generalize the Cauchy's equation to $x \in \mathbb{R}...
14
votes
5answers
422 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 ?
1
vote
1answer
105 views

Cauchy functional equation with non choice

Assume ZF+ not AC. Then how many solutions are there for Cauchy functional equation? Thank you
8
votes
2answers
1k views

Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$

I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I ...
14
votes
3answers
2k views

Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$

Find all functions satisfying $f(2x)=2f'(x)f(x)$ Under given condition, can't we find explicit solutions?
2
votes
2answers
153 views

A non-zero function satisfying $g(x+y)=g(x)g(y)$ must be exponential function

Let $g$ be a non-zero function satisfying $g(x+y)=g(x)g(y)$. Show that the function must be exponential function.
4
votes
1answer
103 views

Functional equation $(1-z)f(x)=f(\frac{1-z}{z}f(xz))$

May you could help me with the following functional equation: $(1-z)f(x)=f(\frac{1-z}{z}f(xz))$. I want to find all function $f:[0,\infty)\rightarrow[0,\infty)$ for $x>0,0<z<1$ Its an ...
6
votes
1answer
385 views

Find all differentiable functions $f$ such that $f\circ f=f$

I want to find all differentiable functions $f:\mathbb R \to \mathbb R$ such that $f\circ f=f$, My attempt since $f$ is differentiable, $f'(f(x))f'(x)=f'(x)$ Now if $f'(x)\neq0$($f'=0$ means ...
1
vote
0answers
142 views

Is there a function $f$ such that $\Gamma (c+x)=\Gamma (c-f\left( x \right) )$?

I was just looking at Euler's reflection formula for Gamma function which states $$\Gamma (1-z)\Gamma (z)=\frac { \pi }{ \sin { (z\pi ) } } $$ but it seems to me that one more reflection formula ...
7
votes
1answer
697 views

Measurable Cauchy Function is Continuous

I found this question here Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable and I want to adapt the proof that t.b had suggested. I don't know the concepts of Baire ...
5
votes
2answers
111 views

Follow on from previous question: Functional Equation - a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. The answer to this has already been posted, but it doesn't explain why ...
1
vote
1answer
205 views

Injectivity of a Function [closed]

Sorry for confusion. I am in the process of solving a functional equation, I need to show injectivity. (By the way i know that it is injective, I'm trying to prove it to myself). Putting $f(x)=f(y)$ ...
7
votes
1answer
495 views

How to find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$?

Can someone please show me how to: Find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$? I've tried substitiuting $x=0,1$. Can't seem to figure it out. The square on the RHS is confusing ...
0
votes
2answers
81 views

Complex Constant and Convergent Power Series

Suppose that the function $f$ is defined by a convergent power series and suppose that $f (z + w) = f (z) f (w)$ for all complex $z$, $w$. (a) Prove directly from this assumption that there is a ...
1
vote
1answer
101 views

If $h(x,c)$ is solution to equation $f'(x)=g(f(x))$, what is the solution for $-f'(x)=g(f(x))$?

Assume that we have differential equation $$f'(x)=g(f(x))$$ and we know that function $h(x,c)$ is the solution (with parameter) to it, then what can be said of the solution for the equation $$-f'(x)=g(...
1
vote
3answers
109 views

Who can solve this ordinary differential equation?

$$f'=f(1-x)$$ This equation appears when I try to solve the eigenvalue problem of an integral equation.
9
votes
2answers
463 views

Functions satisfying $f\left( f(x)^2+f(y) \right)=xf(x)+y$

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f\left( f(x)^2+f(y) \right)=xf(x)+y$ for all real numbers $x$ and $y$. Clearly $f(x)=x$ is a solution, check by substitution. I'm ...
6
votes
3answers
1k views

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)$. I know that there are other questions that are asking the same thing, but I'm trying to figure this out ...
2
votes
2answers
246 views

Easy functional equation

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that: $$f(2f(x)+f(y))=2x+f(y)\qquad \forall x,y \in \mathbb{R}.$$ If you put $x=y=0$, you get $f(3f(0))=f(0)$. What deductions about ...
0
votes
1answer
174 views

Help creating “Halving” Equation

I am not sure if this is the correct site to ask this but here goes, I have a pseudocode for a program algorithm that I am trying to turn into an equation. Say I have a variable X. Now if X is 8, ...
4
votes
1answer
234 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 $\mathbb{R}$-...
2
votes
1answer
103 views

Functional equation on $\mathbb Q_p^\star$

I am trying to find solutions of functional equation $f(xy)=f(x)+f(y)$ for all $x,y\in\mathbb Q_p\setminus\{0\}=\mathbb Q_p^\star$. Where $f:\mathbb Q_p^\star\to\mathbb R$. I know some solutions: 1) $...
11
votes
4answers
389 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?
2
votes
1answer
107 views

functional equation problem in competition

Find all $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that $f(1)=2$ and $f(xy)=f(x)f(y)−f(x+y)+1$. for all $x,y \in \mathbb{Q}$. thank you very much!
7
votes
3answers
308 views

$f(x^2) = 2f(x)$ and $f(x)$ continuous

I ran into a problem recently where I obtained the following constraint on a function. $$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$ and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
0
votes
2answers
38 views

Implicit equation - $G(x,y)=0$

I'm confused about some points in implicit equation ... From my recitation class - $G(x,y)=0$ provides - $y=f(x)$ . And $f'(x)=\frac{dy}{dx}$ and about $G'(x,y)=0$ we use - ...
3
votes
2answers
89 views

Solution to functional equation

I have the following functional equation: $$f(a+b)=f^{n-1}(a)f(b)+f^{n-2}(a)f^{1}(b)+...+f(a)f^{n-1}(b)=\sum_{k=0}^{n-1}f^{n-1-k}(a)f^{k}(b)$$ where $a,b$ are complex and the function $f$ is an ...
0
votes
2answers
545 views

Find all functions that satisfy $f(f(x)+y)=2x+f(f(y)-x)$

Find all the function satisfy: $$f(f(x)+y)=2x+f(f(y)-x), \forall x , y \in \mathbb{R}$$ I have tried that: Let $x:=-x $ we have : $$f(f(y)+x)=2x+f(f(-x)+y) ,(1) $$ Then in $(1)$ $x:=y;y:=x$ we have :...
4
votes
3answers
99 views

An odd function satisfying $g(1-t)+g(1+t)=-t$

I am looking for a continuously diffferentiable odd function $g$ such that $$g(1+t)+g(1-t)=-t$$ for all $t\in\mathbb{R}$. Is this possible?
5
votes
1answer
152 views

Generalization of Cauchy's functional equation

We know that if $f(x+y)=f(x)+f(y)$ and $f$ meets some "reasonable" conditions, then $f$ is linear. I've been considering the following extension: consider the reals under some unknown group operation ...
1
vote
1answer
91 views

“Foldable” functions

Suppose $f:2^X\to X$ satisfies $f(x_1,\dots)=f(f(x_1,x_2),x_3,\dots)$. Min, max and sum are three such examples. I've been calling these functions "foldable" because they bear some similarity to ...
0
votes
1answer
219 views

More general reflection formula for Gamma function

It is know that $\Gamma(z)\, \Gamma{(1-z)}=\pi \csc( \pi z)$. Is there any formula for $\Gamma{(a+z)}\Gamma{(a-z)}$ where $a$ is a rational number, i. e., $a=p/k$ with $p, k$ integers and $z$ is ...
1
vote
1answer
100 views

functional equations

Find all solutions $f:\Bbb R^2 \to \Bbb R$ satisfying $$ f(xu-yv, yu+xv)=f(x, y)f(u, v). $$ Solution of the following equation $$ f(xu+yv, yu-xv)=f(x, y)f(u, v) $$ is known as $$ f(x,y)=m(x^2+y^2), $$...
5
votes
1answer
143 views

Does a function exist with the property $f(-n^2+3n+1)=(f(n))^2+1$?

Let $f:\mathbb{R}\to \mathbb{R}$ be a function which fulfills for every $n \in \mathbb{N}$ $$f(-n^2+3n+1)=(f(n))^2+1$$ Is it possible that such a function exists?
14
votes
4answers
604 views

Polynomial: $p(x) = p(x+3)$.

Determine polynomial $p(x)$ s.t. $p(x) = p(x+3)$. Just by looking at the above equation, it immediately appears that p has got to be some kind of constant function. I thought it might also be a ...
4
votes
2answers
90 views

Functional equation $m(x^y)=m(x)+m(y)$.

Find all functions $m : \mathbb{R}^+ \to \mathbb{R}^+$ such that$$m(x^y)=m(x)+m(y)$$
1
vote
1answer
99 views

Properties of $g$ satisfying $f(x,x)[\nabla_{x}^{2}g(x,y)]_{x=y}+2[\nabla_{x}f(x,y)]_{x=y}\cdot[\nabla_{x}g(x,y)]_{x=y}=0$ for all $f$

Suppose that $g:\mathbb{R}^{2n}\rightarrow\mathbb{R}$ is a function such that: $$f\left(x,x\right)\left[\nabla_{x}^{2}g\left(x,y\right)\right]_{x=y}+2\left[\nabla_{x}f\left(x,y\right)\right]_{x=y}\...
6
votes
1answer
83 views

How can find this function by $x\in \mathbb{Q}^+$

Let $f:\mathbb{Q}^+\to \mathbb{Q}^+,f(x)+f(1/x)=1 $ and $f(2x)=2f(f(x)),x\in \mathbb{Q}^+$, prove that $$f(x)=\dfrac{x}{x+1},x\in \mathbb{Q}^+$$ This Problem from my student.
12
votes
1answer
198 views

How can prove this equation.

if $a+b=c+d=e+f=\dfrac{\pi}{3}$, $\dfrac{\sin{a}}{\sin{b}}\cdot\dfrac{\sin{c}}{\sin{d}}\cdot\dfrac{\sin{e}}{\sin{f}}=1$, Prove that: $\dfrac{\sin{(2a+f)}}{\sin{(2f+a)}}\cdot\dfrac{\sin{(2e+d)}}{\...
11
votes
3answers
295 views

Functions satisfying $\sum_{n=0}^k(-1)^n\binom{k}{k-n}f^{k-n}(x)=0$.

This question was motivated by this one. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Define $f^n(x)=f\circ f\circ\cdot\cdot\cdot\circ f$, $n$ times, $f^0(x)=x$ and $k\geq 2$ a ...
19
votes
1answer
556 views

Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$.

How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying $$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$ are given by $$f(x)=x+a,a\in\mathbb{R}.$$ Any hints are welcome. Thanks.
5
votes
1answer
298 views

$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 \right ),\forall x\in \mathbb{R},m\in \mathbb{Z^+}$

Find all the function $f:\mathbb{R}\rightarrow \mathbb{R}$ sastisfied that $f$ continuous on $\mathbb{R}$ and $$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 \...
6
votes
2answers
167 views

Map that satisfies $f(\lambda x) = \lambda f(x)$ but not $f(x+y) = f(x)+f(y)$

Could you give me example of maps $f:\mathbb R \to \mathbb R$ that satisfy $$ f(\lambda x) = \lambda f(x) \quad \forall x,\lambda \in \mathbb R $$ but not $ f(x+y) = f(x)+f(y) $? Thanks in advance.
4
votes
2answers
230 views

The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$

I came across the following problem that says: The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$ is which of the following? $(1)0\space (2)1 \...
0
votes
2answers
57 views

$f(\vec{x_1}+\vec{x_2})=f(\vec{x_1})f(\vec{x_2})$

What is the general solution to $f(\vec{x_1}+\vec{x_2})=f(\vec{x_1})f(\vec{x_2})$ where $\vec{x}$'s are in discrete vector space $x\in \{n_1\vec{e_1}+n_2\vec{e_2}+n_3\vec{e_3},n_1,n_2,n_3 \in Z\}$?
3
votes
4answers
290 views

Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form $f(x)=ax+b$...
0
votes
0answers
99 views

Interpretation of functional equation of dedekind eta function

It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a ...