Let $f_{n}, n=1,2, \ldots$, be continuous functions on an open interval $(a, b)$. Prove that if the sequence $\left(f_{n}\right)$ converges to $f$ uniformly on $(a, b)$ then the function $f$ is continuous on $(a, b)$.

If instead $\left(f_{n}\right)$ is only known to converge pointwise to $f$ and $f$ is continuous, must $\left(f_{n}\right)$ be uniformly convergent? Justify your answer.

Suppose that a function $f$ has a continuous derivative on $(a, b)$ and let

$$g_{n}(x)=n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)$$

Stating clearly any standard results that you require, show that the functions $g_{n}$ converge uniformly to $f^{\prime}$ on each interval $[\alpha, \beta] \subset(a, b)$.