# 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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### 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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### Finding Value, Related To Functional Equation

$f(x)$ is continuous for $\forall x \in R$ and $f(2x)-f(x)=x^{3}$ (1) $f(x)+f(-x)$ is constant ? (2) $f(0)=0$ ? I don't know how to use the continuity. especially for $f(0)=0$ ?
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### $f: \Bbb N→ \Bbb N$ , $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$

How to find all functions $f: \Bbb N→ \Bbb N$ which satisfy $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$ ($\Bbb N$ is the set of all natural numbers, i.e. positive integers) ?
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### Functions satisfying $\sum_{n=0}^k(-1)^n\binom{k}{k-n}f^{k-n}(x)=0$.

This question was motivated by this one. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Define $f^n(x)=f\circ f\circ\cdot\cdot\cdot\circ f$, $n$ times, $f^0(x)=x$ and $k\geq 2$ a ...
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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.