Convergence of a certain amount

assume $$g_i: [0, 1] R$$ are $$C ^ 1$$ and that there are some $$c> 0$$ as for each $$0 < epsilon , the functions

$$s ( epsilon) _ i$$ $$: = sum_ {k = 0} ^ i { epsilon} ^ k g_k$$

converge uniformly towards a $$C ^ 1$$ a function $$s ( epsilon)$$.

As $$epsilon to 0$$ Is

I) $$s ( epsilon) to s (0)$$ uniformly?

ii) $$s ( epsilon) ' to s (0)'$$ uniformly?