(4) Given the functions $f(x)=\frac{x^{3}+x^{-3}}{5}, g(x)=\frac{x^{3}-x^{-3}}{5}$.
(1) For what real number $m$ does the equation $f(x)=m$ have a solution?
(2) Prove that $f(x)$ is an odd function, and find the intervals of monotonicity for $f(x)$;
(3) Calculate $g(4)-5 f(2) g(2)$ and $g(9)-5 f(3) g(3)$, and from this, generalize an equation involving the functions $f(x)$ and $g(x)$ that holds for all non-zero real numbers $x$, and prove it.
(4) (1) $m \leqslant-\frac{2}{5}$ or $m \geqslant \frac{2}{5}$;
(2) $f(x)$ is monotonically increasing on the intervals $(-\infty,-1]$ and $[1,+\infty)$, and monotonically decreasing on the intervals $[-1,0)$ and $(0,1]$; (3) $g(4)-5 f(2) g(2)=0$, $g(9)-5 f(3) g(3)=0$. An equation that holds for all non-zero real numbers $x$ is $g\left(x^{2}\right)-$ $5 f(x) g(x)=0$