Finding an explicit & bijective function that satisfies the following properties?

Question

Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$.

Question:

Does there exist an explicit and bijective $f:\mathbb{R}\to\mathbb{R}$, where:

1. The function $f$ is measurable
2. The range of $f$ is $\mathbb{R}$
3. For all real $x_1,x_2,y_1,y_2$, where $-\infty<x_1<x_2<\infty$ and $-\infty<y_1<y_2<\infty$: $$\lambda(\left([x_1,x_2]\times[y_1,y_2]\right)\cap\left\{(x,f(x)):x\in\mathbb{R}\right\})>0$$

$\quad\quad\!$ and $$\lambda(\left([x_1,x_2]\times[y_1,y_2]\right)\cap\left\{(x,f(x)):x\in\mathbb{R}\right\})\neq(x_2-x_1)(y_2-y_1)$$

Attempt: I'm not sure how to answer this question but I heard of Conway’s Base-13 function?

I have also asked a similar question here with an answer suggesting that no function satisfies that question.

Answer 1

$\newcommand\R{\mathbb R}\newcommand\la{\lambda}$No. Indeed, let $G:=\{(x,f(x))\colon x\in\R\}$. Then, by the Tonelli theorem, $$\la(G)=\int_\R dx\,\int_{[f(x),f(x)]}dy=\int_\R dx\,0=0.$$ So, for all real $x_1,x_2,y_1,y_2$ such that $-\infty<x_1<x_2<\infty$ and $-\infty<y_1<y_2<\infty$ we have $$\la(([x_1,x_2]\times[y_1,y_2])\cap\{(x,f(x))\colon x\in\R\})=0.$$

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Here we used the fact that the graph $G$ of $f$ is measurable, which follows because $f$ can be uniformly approximated from below and from above by measurable functions taking at most countable many values. Modifying this latter argument slightly, we can do even without the Tonelli theorem.