State and prove Baire's category theorem for complete metric spaces. Give an example to show that it may fail if the metric space is not complete.

Let $f_{n}:[0,1] \rightarrow \mathbb{R}$ be a sequence of continuous functions such that $f_{n}(x)$ converges for all $x \in[0,1]$. Show that if $\epsilon>0$ is fixed we can find an $N \geqslant 0$ and a non-empty open interval $J \subseteq[0,1]$ such that $\left|f_{n}(x)-f_{m}(x)\right| \leqslant \epsilon$ for all $x \in J$ and all $n, m \geqslant N$.

Let $g:[0,1] \rightarrow \mathbb{R}$ be defined by

$$g(x)= \begin{cases}1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational. }\end{cases}$$

Show that we cannot find continuous functions $g_{n}:[0,1] \rightarrow \mathbb{R}$ with $g_{n}(x) \rightarrow g(x)$ for each $x \in[0,1]$ as $n \rightarrow \infty .$

Define a sequence of continuous functions $h_{n}:[0,1] \rightarrow \mathbb{R}$ and a discontinuous function $h:[0,1] \rightarrow \mathbb{R}$ with $h_{n}(x) \rightarrow h(x)$ for each $x \in[0,1]$ as $n \rightarrow \infty$.