State and prove the Baire Category Theorem. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. For $x \in \mathbb{R}$, define

$$\omega_{f}(x)=\inf _{\delta>0} \sup _{\substack{|y-x| \leqslant \delta \\\left|y^{\prime}-x\right| \leqslant \delta}}\left|f(y)-f\left(y^{\prime}\right)\right|$$

Show that $f$ is continuous at $x$ if and only if $\omega_{f}(x)=0$.

Show that for any $\epsilon>0$ the set $\left\{x \in \mathbb{R}: \omega_{f}(x)<\epsilon\right\}$ is open.

Hence show that the set of points at which $f$ is continuous cannot be precisely the set $\mathbb{Q}$ of rationals.