A dense bijection on $\mathbb{R}$ ?

Are there any functions satisfying the following properties ?

$(i) \hspace{7mm} f: \mathbb{R}\longrightarrow\mathbb{R} \ \text{is a bijection} \\ (ii) \hspace{5.7mm} \{(x,f(x)), \ x\in\mathbb{R}\} \ \text{is dense in} \ \mathbb{R}^2$

Note: here dense means dense with respect to the classics norms on $\mathbb{R}^2$ such as $\lVert \cdot \rVert_1$ or $\lVert \cdot \rVert_2$ (which are equivalent).