# sets where sequence of continuous functions diverge

Related Exercise from Rudin’s Real and Complex Analysis.

Exercise 5.20

• (a)

Does there exist a sequence of continuous positive functions $f_{n}$ on $\mathbb{R}^{1}$ such that $\{f_{n}(x)\}$ is unbounded if and only if $x$ is rational?

• (b)

Replace “rational” by irrational in (a) and answer the resulting question.

• (c)

Replace “$\{f_{n}(x)\}$ is unbounded” by “$f_{n}(x)\to\infty$ as $n\to\infty$” and answer the resulting analogues of (a) and (b).

Solution: The answer to (a) is negative. This by showing that the subset of points where such sequence is unbounded must be $G_{\delta}$. But the rationals cannot be such, since in $\mathbb{R}$ dense $G_{\delta}$ sets must be of second category.