Even, odd and periodic functions

A function $f(x)$ is defined on a symmetric interval $(-l,l)$, and if

then $f(x)$ is called an even function; if

then $f(x)$ is called an odd function.

Determine which of the following given functions are even and which are odd:

Prove that any function $f(x)$ defined on a symmetric interval $(-l,l)$ can be expressed as the sum of an even function and an odd function.

If there exists a number $T>0$ (the period of the function — in the broad sense) such that for all considered independent variables $x$ the equality

holds, then the function $f(x)$ is called a periodic function.

Determine which of the following given functions are periodic functions, and find their smallest periods. Let:

Prove: for the Dirichlet function

any rational number is its period.

Prove that the sum and product of two periodic functions defined on a common set, whose periods are commensurable, are also periodic functions.

Prove: if for the function $f(x) (-\infty <x<+\infty )$ the equality

(where $k$ and $T$ are positive constants) holds, then

(where $a$ is a constant greater than zero, and $\varphi (x)$ is a function with period $T$).

A function $f(x)$ is defined on the domain of positive numbers $x>0$. Extend $f(x)$ to the domain of negative numbers $x<0$ so that the resulting function is: (1) even; (2) odd. Let

Determine about which vertical axis the graph of each of the following functions is symmetric:

Determine the center of symmetry of the graph of each of the following functions:

Sketch the graphs of the periodic functions:

Prove: if the graph of the function $y=f(x)$ ($-\infty <x<+\infty$) is symmetric with respect to the two vertical lines $x=a$ and $x=b$ ($b>a$), then the function $f(x)$ is periodic.

Prove: if the graph of the function $y=f(x)$ ($-\infty <x<+\infty$) is symmetric with respect to the two points $A(a,y_{0})$ and $B(b,y_{1})$ ($b>a$), then the function $f(x)$ is the sum of a linear function and a periodic function. In particular, if $y_{0}=y_{1}$, then $f(x)$ is a periodic function.

Prove: if the graph of the function $y=f(x)$ ($-\infty <x<+\infty$) is symmetric with respect to the point $A(a,y_{0})$ and the line $x=b$ ($b \neq a$), then the function $f(x)$ is periodic.

Given $f(x+1)=2f(x)$ and, for $0 \leqslant x \leqslant 1$, $f(x)=x(1-x)$, sketch the graph of the function

Given $f(x+\pi )=f(x)+\sin x$, and for $0 \leqslant x \leqslant \pi$, $f(x)=0$, sketch the graph of the function

If $\varphi (x)$ and $\phi (x)$ are continuous periodic functions defined for $-\infty <x<+\infty$, and

prove that $\varphi (x)\equiv \phi (x)$.

Prove that the derivative of a differentiable even function is an odd function, and the derivative of a differentiable odd function is an even function.

Give a geometric interpretation of this fact.

Prove that the derivative of a differentiable periodic function is again a periodic function with the same period.