Questions tagged [functional-equations]
The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).
4,190 questions
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Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is bounded above and $f(xf(y))+yf(x)=xf(y)+f(xy)$
Find all functions $f : \mathbb R \to \mathbb R$ such that: $f (x f ( y ))+y f ( x) = x f ( y ) + f ( x y )$, $\forall\ x , y \in \mathbb{R}$
and
b) $\exists M \in \mathbb R$ such that $f(x)<M$ ...
- 105
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$f(x) + f(t) = f(x+t)$ then $f(x)=kx$ [duplicate]
I wanna to prove
If for all value of $x$ and $x'$ in real numbers we have;
$$f(x) + f(x') = f(x+x')$$
Then
$f(x) = Kx.$
I have not any counterexample for this but I can't prove it
Thank you for your ...
2
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1
answer
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Generalization of Cauchy's functional equation. What are the general solutions, $f$?
Consider a function $f : (0,\infty) \to (0,\infty)$ satisfying the identity
$$
f(x^a y^b) \;=\; f(x)^{1/a}\, f(y)^{1/b}
\qquad\text{for all } x,y>0 \text{ and all real } a,b\neq 0.
$$
This can be ...
- 1,209
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Additive decomposition of function [closed]
If $f$ is a function such it is commutative, continuous and homogeneous and that for any positive variables $a,b$ and $c$, $f(f(a, b), c) = f(f(a, c), b)$ then prove that there must exist a non ...
4
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Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$
Problem: Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$.
My approach: I have got $f(0)=0,1$. Taking $f(0)=1$, and setting $...
- 499
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Is there an easy way to know if a rational function is an n-th (compositional) iteration of a power series with indices in $\mathbb{Z}$?
I am asking this question mainly to probe the knowledge of people already familiar with this problem, otherwise I would advise caution to the unfamiliar trying to use computation, this can be really ...
- 2,911
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Are there any non-trivial functions satisfying $tf(x)=f\left(x+\frac{1}{t}-1\right)$?
I'm looking for a function, $f$, which satisfies
\begin{align}
tf(x)=f\left(x+\frac{1}{t}-1\right);\quad f(1)=1.
\end{align}
My attempt:
Let $t\rightarrow 1/(1+\log t)$ and $x\rightarrow \log x$, ...
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Yet another nice functional inequality [duplicate]
Need to find out all the functions $f \colon \mathbb{R} \to \mathbb{R}$ such that
$$f(x+y) \leq f(xy).$$
Since $f(x) \leq f(0)$ for every $x \in \mathbb{R}$ and $f(0) \leq f(-x^2)$, it follows that $f(...
- 629
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votes
1
answer
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Characterization of holomorphic function whose n-fold composition is the identity
Suppose $f$ is a holomorphic function defined on a neighborhood of $0$, such that $f(0) = 0$. Let $f^n$ denote the $n$-fold composition of $f$ with itself. If $f^n$ is the identity for some natural ...
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Classification of monotone associative laws on a bounded interval with Möbius translations fixing the boundary (unique $\tanh$ linearizer)
Let $\kappa>0$ and $I_\kappa=\bigl(-\tfrac{1}{\kappa},\tfrac{1}{\kappa}\bigr)$.
Consider a binary operation $\oplus:I_\kappa^2\to I_\kappa$ with: (A1) $\oplus$ is continuous, strictly increasing in ...
7
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Find all the functions $f \colon \mathbf R \to \mathbf R$, such that $f(x+\frac{1}{y})+f(y+\frac{1}{x})=2f(xy)$ for all $x,y \in \mathbf R$.
the problem
Find all the functions $f \colon \mathbf R \to \mathbf R$, such that $$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{x}\right)=2f(xy)$$ for all $x,y \in \mathbf R$.
my idea
Plugging in ...
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A strange functional equation
Suppose that an unbounded function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(a+b)=f(a)+f(b)$ whenever $f(a+b)$ is the maximum of $f(0), f(1), \dots, f(a+b)$, where $a, b\in \mathbb{N}$. Show ...
- 111
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Is there a name for this type of function?
Suppose a function $f(x)$ is defined on a domain $D$ where $D$ is a subset of $\mathbb{R}$ and $D$ has the property that $x\in{D}$ iff $\frac{1}{x}\in{D}$.
Furthermore, suppose $f$ has the following ...
- 1,563
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Interesting case of a system of functional equations
Building upon this question, I would like to extend the unanswered question in the comments.
That is, do there exist $g,h: \mathbb{N} \rightarrow \mathbb{N}, y \in \mathbb{N}$, such that the following ...
2
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The existence of an additive function for which the set of its zeros is exactly the set of all rational numbers
I've recently read about discontinuous additive function (i.e. discontinuous solutions of the Cauchy functional equation $f(x+y)=f(x)+f(y)$, $x,y\in\mathbb{R}$) and the following problem came to my ...
