Question 79
Find all functions $f, g: [0; 1] \to [0; 1]$, satisfying the following conditions:
$$
\begin{cases}
\{f(x)\} \sin^2 x + \{x\} \cos f(x) \cos x = f(x), \\
f(f(x)) = f(x),
\end{cases}
$$
for all real $x$.
(Here $\{y\}$ stands for the fractional part of $y$.)
$$
\begin{cases}
\{f(x)\} \sin^2 x + \{x\} \cos f(x) \cos x = f(x), \\
f(f(x)) = f(x),
\end{cases}
$$
for all real $x$.
(Here $\{y\}$ stands for the fractional part of $y$.)