This is problem 13 from page 264 of Pugh's Real Mathematical Analysis. Let $(f_n)$ be a sequence of functions $\mathbb{R}\mapsto\mathbb{R}$ such that if $K \subset \mathbb{R}$, $K$ compact, then the restricted sequence $(f_n|_K)$ is pointwise bounded and pointwise equicontinuous.
a) Does it follow that there is a subsequence $(f_n)$ that converges to a continuous limit function $\mathbb{R}\mapsto\mathbb{R}$?
b) What about uniform convergence?
I really don't know how to approach this problem. Hints or a solution would be appreciated.