Let
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
be a function from the set of real numbers
![\mathbb{R}](/media/m/1/4/0/140a3cd0f5aa77f0f229f3ae2e64c0a6.png)
into itself such for all
![x \in \mathbb{R},](/media/m/5/3/a/53ac72d7a7cec7df010582390195c74b.png)
we have
![|f(x)| \leq 1](/media/m/c/4/1/c41225e37d2718620ce3b55188f8a352.png)
and
Prove that
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
is a periodic function (that is, there exists a non-zero real number
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
such
![f(x+c) = f(x)](/media/m/e/e/e/eeed4e45a95bcdc710259e7a84ac863b.png)
for all
![x \in \mathbb{R}](/media/m/c/1/6/c162a96a3e70076e1031361250564464.png)
).
%V0
Let $f$ be a function from the set of real numbers $\mathbb{R}$ into itself such for all $x \in \mathbb{R},$ we have $|f(x)| \leq 1$ and
$$f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac{1}{6} \right) + f \left( x + \frac{1}{7} \right).$$
Prove that $f$ is a periodic function (that is, there exists a non-zero real number $c$ such $f(x+c) = f(x)$ for all $x \in \mathbb{R}$).