Functional Equations

Describe all functions $f \in C(X)$ satisfying the equation $f(2 x)=f(x)$ for every $x \in X$, in the following cases:

Let the function $f: \mathbb {R} \rightarrow \mathbb {R}$ satisfy the equation $f(x+y)= =f(x)+f(y)(x, y \in \mathbb {R})$. Prove that

Describe all monotone functions $f: \mathbb {R}_{+} \rightarrow \mathbb {R}$ satisfying the equation $f(x y)=f(x)+f(y)$ for all $x, y>0$.

Describe all functions $f \in C(\mathbb {R})$ satisfying the equation $f(x y)=x f(y)+y f(x)$ for any $x, y \in \mathbb {R}$.

Find all functions $f: \mathbb {R} \rightarrow \mathbb {R}$ satisfying the system of equations

Let $n \in \mathbb {N}, n>1$. Describe all functions $f: \mathbb {R} \rightarrow \mathbb {R}$ such that $f\left(x+y^{n}\right)=f(x)+(f(y))^{n}$ for any $x, y \in \mathbb {R}$.

Let the function $f \in C(\mathbb {R})$, for some $\sigma >0$, satisfy the condition $\left|f(x)+f(y)-f(x+y)\right| \leqslant \sigma$ for any $x, y \in \mathbb {R}$. Prove that $f$ is representable as the sum of a linear function and a function not exceeding $\sigma$ in absolute value.

Let the function $f \in C(\mathbb {R})$ satisfy the condition $f(x)+f(y)= =f\left(\sqrt{x^{2}+y^{2}}\right)$ for any $x, y \in \mathbb {R}$. Prove that $f(x)=a x^{2}$ for any $x \in \mathbb {R}$, where $a=f(1)$.

Let the function $f \in C(\mathbb {R})$ satisfy the condition $f \not\equiv 0$, $f(x) f(y)=f\left(\sqrt{x^{2}+y^{2}}\right)$ for all $x, y \in \mathbb {R}$. Prove that $f(x)=e^{a x^{2}}$ for all $x \in \mathbb {R}$ ($a$ is a fixed number).

Generalize this result to the case when

for $x, y \in \mathbb {R}$ and some $p>0$.

Let $a>b>0$. Describe all functions $f$ from $C\left(\mathbb {R}_{+}\right)$ such that the difference $f(a x)-f(b x)$ does not depend on $x \in \mathbb {R}_{+}$.

Find a continuous strictly increasing function $f: \mathbb {R} \rightarrow \mathbb {R}$ such that $f\left(x^{2}\right)=f^{4}(x)$ for all $x \in \mathbb {R}$.

Let $\bar{B}=\left\{ z \in \mathbb {C}\left| \right| z \mid \leqslant 1\right\}$. Find all continuous bijective mappings $f$ of the disk $\bar{B}$ onto itself satisfying the condition $f\left(z^{2}\right)=f^{2}(z)$ for all $z \in \bar{B}$.

Prove that there exists a unique monotonic function $f:\left[0,1\right] \rightarrow \mathbb {R}$ satisfying the conditions

for all $x \in [0,1]$.

Find all functions $f \in C^{2}\left(\mathbb {R}^{2}\right)$ satisfying, for all $x, y, z \in \mathbb {R}$, the conditions

For which continuous functions $f: \mathbb {R} \rightarrow \mathbb {R}$ does there exist a function $g: \mathbb {R}^{2} \rightarrow \mathbb {R}$ such that $f(x y)=g(x, f(y))$ for all $x, y \in \mathbb {R}$?

Describe all continuous functions $f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ satisfying the conditions:

1. $\min \left\{ x_{1}, x_{2}, \ldots , x_{n}\right\} \leqslant f(x) \leqslant \max \left\{ x_{1}, x_{2}, \ldots , x_{n}\right\} \quad$ for all $x=\left(x_{1}, x_{2}, \ldots , x_{n}\right) \in \mathbb {R}$;

2. if $f(x)<f(y)$, then $f(x+z)<f(y+z)$ for any $z \in \mathbb {R}^{n}$.

Which of these functions are symmetric (the value $f(x)$ is preserved under any permutations of the coordinates $x_{1}, \ldots , x_{n}$)?

Find the general form of the characters in the groups:

Find the general form of the continuous characters on the groups (additive)

We will call a continuous function $f: \mathbb {R} \rightarrow S^{1}$ a character of the second degree if it satisfies the functional equation ($x, y, z \in \mathbb {R}$)

Prove that

Describe all continuous homomorphisms $\varphi$ of the group $G$ into the group $G^{\prime }$ in the following cases (using the notation introduced in problem 5.19):