I was thinking about derivative of infinite sum of functions, i.e.
$f(x) = \sum_{i = 0}^\infty g_i(x)$
$g(x)$ is continuous in domain of $f$
Because if (f+g)'(x) = f'(x) + g'(x) then \left(\sum\limits_{i = 0}^{\infty} g_i(x)\right)' = \sum\limits_{i = 0}^{\infty} g_i'(x) isn't it?