Exercise 22, § 5

a) Let $X$ be a Baire space, $Y$ a metric space, $(f_n)$ a sequence of continuous mappings of $X$ into $Y$ such that, for each $x \in X$, the sequence $(f_n(x))$ has a limit $f(x)$ in $Y$. Then the set of all points $x \in X$ at which $f$ is not continuous is meagre. [Prove that, for each integer $n > 0$, the set of all points $x \in X$ at which the oscillation $\omega(x; f)$ of $f$ is $\leq 1/n$ contains a dense open set; for this, if $G_p$ denotes the set of $x \in X$ for which the distance from $f_p(x)$ to $f_q(x)$ is $\leq 1/2n$ for all $q \geq p$, show that the union of the interiors of the sets $G_p$ is a dense open set.]

b) Let $K$ be the compact interval $[0, 1]$ in $\mathbf{R}$. Give an example of a sequence of continuous mappings $f_n$ of $K$ into the compact space $K^K$ such that $\lim_{n \to \infty} f_n(x) = f(x)$ exists for each $x \in K$, but such that $f$ is not continuous at any point of $K$.