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 <c $, 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?