The Dirichlet function is defined by $f(x)=\begin{cases} 1 &\text{ if } x\in \mathbb{Q}\\0 &\text{ if } x\notin \mathbb{Q}.\end{cases}$ Let $g(x)=\begin{cases}0 &\text{ if } x\in \mathbb{Q}\\ 1 &\text{ if } x\notin \mathbb{Q} \end{cases}$ be its evil twin.
(1) Prove that $f$ is discontinuous at every $x\in \mathbb{R}$.
(2) Prove that $g$ is continuous at exactly one point $x_1\in \mathbb{R}$.
(3) Prove that $f+ g$ is continuous at every $x\in \mathbb{R}$.
Definition (Continuous at a point): A function $f: D \longrightarrow \mathbb{R}$ is continuous at $X_0\in D$ if for every $\epsilon > 0$ there exists some $\delta > 0$ such that $|f(x) - f(x_0)| < \epsilon$ whenever $|x-x_0|< \delta$.
Response: I don't have any work yet, hints and answers would be extremely helpful.