# Tagged Questions

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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### Functional equation with only two solutions?

Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation: $$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$ Unfortunately, I was able to show only ...
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### Functional equation with sqrt solutions [closed]

Let $f:(0,\infty)\to(0,\infty)$ so that for all $x,y\in(0,\infty)$ we have $$f\left(\frac {x} {f(y)}\right)=\frac {x} {f(x\sqrt y)}.$$ Find function $f$.
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### Functional equation: $f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$

Find an algebraic function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$$ and $$f(1)=1$$ for all $x\in\Bbb N$ I allready know two things: $f(p^k)=p$ where $p$ is prime ...
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### Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that $f(x)=f(x^y)$

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(x)=f(x^y)$$ for all $x,y\in\mathbb{N}$. I'm not intrested in the trivial solution $f(x)=k$, where $k\in\mathbb{N}$.
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### System of linear Volterra integral equation

Consider the Volterra integral equation $$f(t) = g(t) +\int_0^t K(t,s) f(s) ds$$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...
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### if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.

Let $f$: $\Bbb R$ $\rightarrow$ $\Bbb R$ be a non-constant function such that $f(a + b) = f(a)f(b)$ for all real numbers $a$ and $b$. Prove that if $f(x + y) = f(x)f(y)$ is continuous, then it has ...
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### Show that $f$ is continuous and hence of the form $f(x)=cx$ [duplicate]

Let $f:\Bbb R\to \Bbb R$ be additive i.e. satisfies $f(x+y)=f(x)+f(y)$ for all $x$, $y$. and Lebesgue measurable. Show that $f$ is continuous and hence of the form $f(x)=cx$. In order to show ...
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### Find all $f(x)$ such that $f(gcd(x,y))=gcd(f(x),f(y))$
How does one find all $f:\mathbb {Z} \rightarrow \mathbb {Z}$ that satisfies the following: $$f(gcd(x,y))=gcd(f(x),f(y))$$ I had suspected that there would be some results concerning this functional ...