Let $A \subset \mathbb{R}$. What does it mean to say that a sequence of real-valued functions on $A$ is uniformly convergent?

(i) If a sequence $\left(f_{n}\right)$ of real-valued functions on $A$ converges uniformly to $f$, and each $f_{n}$ is continuous, must $f$ also be continuous?

(ii) Let $f_{n}(x)=e^{-n x}$. Does the sequence $\left(f_{n}\right)$ converge uniformly on $[0,1]$ ?

(iii) If a sequence $\left(f_{n}\right)$ of real-valued functions on $[-1,1]$ converges uniformly to $f$, and each $f_{n}$ is differentiable, must $f$ also be differentiable?

Give a proof or counterexample in each case.