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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solving functional equation $f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$ for all real numbers [duplicate]

The functional equation to be solved is $f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$. Domain: Reals, Codomain: Reals.I found about 4 possible solutions to the equation but ran into a fundamental problem with ...
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Tricky probability problem

I am having trouble with proving the following assertion: $X,Y$ are i.i.d. with mean $0$ and variance $1$. If $X+Y$ and $X-Y$ are independent then $X,Y$ are normally distributed. Should I be ...
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If $f\circ f\circ f=id$, then $f=id$ [duplicate]

Let $f$ a continuous function on all $\mathbb R$. How can I prove that if $f\circ f\circ f=id$, then $f=id$ ? I really have no idea.
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Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
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Function preserving exponentiation [duplicate]

I'm wondering what kind of function preserves exponentiation, i.e., what is an $f$ such that $f(a^b)=f(a)^{f(b)}$?
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Concerning nonlinear functional equations

There's a problem I've been working on for awhile that involves some hefty functional equations. For example, I may have something along the lines of $$f(x)f(x) =x+1+f(x+1)$$ I've tried several ...