Function term series? ∑an(x) converges to the closed interval [a, b]

The derivative function an'(x) of an(x) is continuous in the closed interval [a, b].

Series? ∑an'(x)？ Uniformly converge to the closed interval [a, b]

Sum function S(x) satisfying the above conditions? =∑an(x)？ Derivable on the closed interval [a, b]

It can also be deduced item by item.

Moreover, the derivative function S'(x) of the sum function is also continuous in the closed interval [a, b].