# 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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### continuous (on 3, 4 and 5) f is constant, if $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is continuous on 3, 4 and 5, such that $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$. Show that f is constant. I don't know what to do ...
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### How to solve this functional equation: $f(1-f(x))=1-x^{9}, f(1)=0$

I have managed to guess one solution of this function : $f(x)=1-x^{3}$, but I have no idea how to prove it unique, or get other solutions. If this is not solvable, how can you prove this function ...
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### The functional equation and differentiability

Find all functions $f: \mathbb R\rightarrow \mathbb R$, at the same time satisfying the following two conditions: a) $f (x + yf (x)) = f (x) f (y)$ b) the function $f$ can be represented in ...
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### Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations

Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations: $f(x)g(y) = x+y$ and $f(x) + g(y) = xy$. I think we should ...
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### If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Show that if $f$ is continuous, then $f(x) = cx$ for all $x \in \mathbb{R}.$ I find it hard ...
### Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$
Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$. We know ...
### Can I prove that $f(x,y)$ can be written as $g(x+y)$ under certain conditions.
I have $f(x,y):R^2\rightarrow R$. I know $f(x,y)=f(y,x)$ and $f(x+d,y)=f(x,y+d)$. Can I prove that I can express $f(x,y)$ as $g(x+y)$. This is where I got: $f(x+d,y)=f(x,y+d)$, I plug in $x=0$ Gives ...