# 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).

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### A function whose antiderivative equals its inverse.

Does there exist a continuous function $F$ satisfying the property \begin{align} F\left(\int^x_0 F(s)\ ds\right) = x \end{align} If yes, then is the solution unique? As stated, the question is ...
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### $f(f(x)f(y))+f(x+y)=f(xy)$

Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$: $$f(f(x)f(y))+f(x+y)=f(xy).$$ It was six hours ago in IMO 2017 (problem 2). I tried the standard ...
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### A possible converse to the Cayley-Hamilton theorem?

Happy new year MSE! During my holiday vacation I had an interesting idea! The Cayley-Hamilton theorem states that if $f:\mathbb C^n\to\mathbb C^n$ is a linear function, then it is a root of its own ...
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### 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?
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### Proving that $f(n)=n$ if $f(n+1)>f(f(n))$

How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$?
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### Class of integrals: $I(a)=\int_0^\infty \frac{dx}{e^x+ax}$

I'm investigating integrals in the form $$I(a):=\int_0^\infty \frac{dx}{e^x+ax}$$ So far, I haven't been able to find any special values other than $I(0)=1$, and I've only managed to evaluate these ...
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### Solving the infinite radical $\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+…}}}}$

$$\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+\cdots}}}}$$ This is a modification on the well-known Ramanujan infinite radical, $\sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}}$, except it cannot be solved by the ...
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### Prove that function is constant

Prove that a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies $$f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$$ is a constant function. This is my solution: constant function have derivative $0$ ...
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### A functional equation with no solution

Let $f:\mathbb{R}\to (0,\infty)$ be a differentiable function satisfying $$f(f(x))=f^\prime(x)$$for each $x$. Show no such function exists. I got this problem in an exam. I haven't done anything ...
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### Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$

I was trying to find functions $f:(0,+\infty)\to(0,+\infty)$ satisfying the following functional equation $$f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$$ The problem is that I can't find here any reasonable ...
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### 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.
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### Find $f(x)$ where $f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $f(x)+f\left(\frac{1-x}x\right)=x$ ?
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### If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t

Let $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$. I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as $g(xy) = g(x) + g(y)$ ...
### Find all functions $f: \mathbb N \rightarrow \mathbb N$ such that $f(n!)=f(n)!$
Find all functions $f: \mathbb N \rightarrow \mathbb N$ (where $\mathbb N$ is the set of positive integers) such that $f(n!)=f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$...
Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...