Suppose $ g_i: [0, 1] \to \Bbb R$, $i\in\Bbb N$, are $C^1$ functions and that there is some $c > 0$ such that for every $0 < \epsilon < c$, the functions $$ s(\epsilon)_i := \sum_{k=0}^i {\epsilon}^k g_k $$ converge uniformly to a $C^1$ function $s(\epsilon)$.

As $\epsilon \to 0$ does

i) $s(\epsilon) \to s(0)$ uniformly?

ii) $s(\epsilon)’ \to s(0)’$ uniformly?