# 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 derivative of an integral of the function itself

I have the following $$\frac{d}{dn(x)} \int_{x \in \cal{R}^3} {n(x) dx}$$ I know that this additional relationship holds $$\int_{x \in \cal{R}^3}{n(x) dx} = N$$ where N is a constant. My ...
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### How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable

Suppose $f(x)$ is integrable in any bounded interval on $\mathbb R$, and it satisfies the equation $f(x+y)=f(x)+f(y)$ on $\mathbb R$. How to prove $f(x)=ax$?
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### Is there a nontrivial solution to $f(f(f(x)))=-8x$?

Let $f:\mathbb R\to\mathbb R$ be a continuous function such that $f(f(f(x)))=-8x$. Must we have $f(x)=-2x$? I can prove this if I assume that $f$ is continuously differentiable everywhere, but is that ...
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### $f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?

A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the ...
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### Solve $f(f(n))=n!$

What am I doing wrong here: ( n!=factorial ) Find $f(n)$ such that $f(f(n))=n!$ $$f(f(f(n)))=f(n)!=f(n!).$$ So $f(n)=n!$ is a solution, but it does not satisfy the original equation except for ...
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### If $f\circ f$ is smooth, is the monotonic function $f$ smooth?

Let $f:\mathbb R\to\mathbb R$ be continuous and monotonic and assume that $f\circ f$ is analytic. Is $f$ necessarily continuously differentiable/smooth/analytic? My question arose from this: Inspired ...
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### Trying to solve the equation involving floor function

I am trying to solve the following equation: Floor[logx+1]+x=11 Where Floor function returns the greatest integer smaller than the value in bracket. e.g. Floor[3.3] = 3 And the logarithm is to the ...
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### Is this a functional equation of a line?

Let $f$ be a real-valued function satisfying the functional equation $$f(x)=f(x+y)+f(x+z)-f(x+y+z)$$ for all $x,y,z\in\mathbb{R}$. Is it true that $f$ must be the equation of a line, with no ...
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### a continuous function satisfying $f(f(f(x)))=-x$ other than $f(x)=-x$

My question is about existence of a non-trivial solution of the functional equation $f(f(f(x)))=-x$ where $f$ is a continuous function defined on $\mathbb{R}$. Also, what about the general one ...
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### Help remembering a Putnam Problem

I recall that there was a Putnam problem which went something like this: Find all real functions satisfying $$f(s^2+f(t)) = t+f(s)^2$$ for all $s,t \in \mathbb{R}$. There was a cool trick to ...
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### Strange functional equation ( hyperfunctions? )

Can we solve this strange functional equation? $$f(x+i\epsilon)-f(x-i\epsilon) = g(x)$$ I believe that the solution is the Hilbert (finite part) transform of the function g(x) however I do not ...
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### designing an equation that compares two values and returns a probability

Given two values, I'm trying to come up with a formula that will return 50% if both values are equal, 25% if the first value is half the second, 75% if the second is half the first. In other words: ...
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### $f(x+f(x+y))=f(x-y)+f(x)^2 \quad \forall x,y\in \mathbb R$

We have to find all functions $f\colon \mathbb R\to\mathbb R$ such that f: $$\forall x,y\in \mathbb R \quad f(x+f(x+y))=f(x-y)+f(x)^2.$$ Could somebody help me solve this problem? Thank you.
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### Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...
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### Starting out with functional equations

I am thinking of starting learning about various functional equations and ways to solve them, any help as to which books could be of help to me? I have some knowledge about some basic functional ...
### $f(x+f(y))=f(x)+y^n$
Here is the problem: Fix $n\in\mathbb{N}$. Find all monotonic solutions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+y^n$. I've tried to show that $f(0)=0$ and derive some ...