Periodic functions

Prove that a periodic function, continuous on the entire number line and not identically constant, has a smallest positive period. Give an example of a periodic function, defined on the entire number line and not identically constant, that has no smallest positive period.

Let $f$ and $g$ be continuous functions, periodic with period $T$, and $\lim_{x \rightarrow \infty }(f(x)-g(x)) = 0$. Prove that $f = g$.

A continuous function $f$ has two incommensurable periods, $T_{1}$ and $T_{2}$ (i.e., $T_{1} / T_{2}$ is an irrational number). Prove that $f =$ const.

Construct functions $f$ and $g$, continuous on $\mathbb {R}$, such that for any $T > 0$ the functions $f(x+T)$ and $g(x)$ are different, but for every $\varepsilon > 0$ there is $T_{\varepsilon } > 0$ such that for any $x \in \mathbb {R}$ the inequality holds

The function $f$ is continuous and periodic with period $T$. Prove that there is a point $x_{0}$ such that

Prove that the function $y = x-\mathbb {E}\left[x\right], x \in \mathbb {R}$, is periodic, and find its smallest positive period.

Prove that the Dirichlet function

is periodic, and that every nonzero rational number is a period of it, while no irrational number is a period.

Find the smallest positive period of the function:

Prove that the function is not periodic:

Prove that if the function $y = f(x)$ is periodic with period $T$, then the function $y = f(a x+b), a \neq 0$, is periodic with period $T / a$.

Prove that the given function is periodic, and find its smallest positive period:

Prove that the given function is not periodic:

Prove that if the ratio of the periods of the periodic functions $f$ and $g$ is a rational number, then the functions $f+g$ and $f g$ are periodic.

Find the smallest positive period of the function:

For which $a$ and $b$ $(a b \neq 0)$ is the function $y = a x-E(b x+c)$ periodic, and what is its smallest positive period?

Give examples of nonperiodic functions $f$ and $g$ such that the functions:

Give examples of a periodic function $f$ and a nonperiodic function $g$ such that the functions:

Does there exist a function for which every irrational number is a period, while no rational number is?

The graph of the function $y = f(x), x \in \mathbb {R}$, is symmetric with respect to each of the lines $x = a$ and $x = b, a \neq b$. Prove that $y = f(x)$ is a periodic function, and find its period.

The graph of the function $y = f(x), x \in \mathbb {R}$, is symmetric with respect to the point $A(a ; b)$ and the line $x = c(c \neq a)$. Prove that $f(x)$ is a periodic function, and find its period.

Prove that the function $f$ is periodic if there exists $T \neq 0$ such that for any $x \in D(f)$, $x+T \in D(f), x-T \in D(f)$, and one of the following conditions holds:

Let the function $g$ be inverse to itself, and let the composition $g \circ f$ be defined. Suppose there exists $T \neq 0$ such that for any $x \in D(f)$ the conditions hold

Prove that $f$ is a periodic function, and find its period.

For the function $f$ there exist numbers $T \neq 0$ and $a$ such that for any $x \in D(f)$ the following holds

and one of the equalities holds:

For the function $f$ there exist a number $T \neq 0$ and a polynomial $Q_{n}(x)$ of degree $n$ such that for any $x \in D(f)$ the following holds

and

Prove that there exists a polynomial $P_{n+1}(x)$ of degree $n+1$ such that

where $\varphi (x)$ is a function periodic with period $T$.

For the function $f$ there exist numbers $T \neq 0$ and $k > 0$ such that for any $x \in D(f)$ the following holds

and

Prove that there exists a number $a > 0$ such that

where $\varphi (x)$ is a periodic function.