Space-filling (Peano) curves

Let $f$ and $g$ be two continuous functions with bounded variation on $[a,b]$. Prove that the set $(f(x),g(x))$, $x\in [a,b]$, cannot fill the square. Is this true without the requirement of bounded variation?

Let $X$ be an arcwise connected compact metric space. Construct a continuous mapping of $[0,1]$ onto $X$.

Construct a continuous mapping of $[0,1]$ onto the unit square.

Construct a continuous mapping of the segment $0 \leqslant x \leqslant 1$ onto the square $0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1$ (a Peano curve).

On the interval $\left[0,1\right]$ define functions $\varphi$ and $\psi$ as follows. Let $t \in [0,1], t=\sum 3^{-k} \alpha_{k}$, where $\alpha_{k} \in \left\{ 0,1,2\right\}$. Then $\varphi (t)=\sum 3^{-k} \beta_{k}$, where $\beta_{1}=\alpha_{1}, \beta_{2}=\alpha_{3}$, if $\alpha_{2}$ is an even number, $\beta_{2}=2-\alpha_{3}$, if $\alpha_{2}$ is an odd number, $\beta_{n}=\alpha_{2 n-1}$, if $\alpha_{2}+\alpha_{4}+\ldots +\alpha_{2 n-2}$ is an even number, and $\beta_{n}=2-\alpha_{2 n-1}$, if $\alpha_{2}+\alpha_{4}+\ldots +\alpha_{2 n-2}$ is an odd number ($n \geqslant 2$). Similarly, $\psi (t)=\sum 3^{-k} \gamma_{k}$, where $\gamma_{1}=\alpha_{2}$, if $\alpha_{1}$ is an even number, $\gamma_{1}=2-\alpha_{2}$, if $\alpha_{1}$ is an odd number, $\gamma_{n}=\alpha_{2 n}$, if $\alpha_{1}+\alpha_{3}+\ldots +\alpha_{2 n-1}$ is an even number, and $\gamma_{n}=2-\alpha_{2 n}$, if $\alpha_{1}+\alpha_{3}+\ldots +\alpha_{2 n-1}$ is an odd number.

A continuous map of an interval onto a square is called a Peano curve. Let $u$ and $v$ be the coordinate functions of an arbitrary Peano curve defined on the interval $\left[0,1\right]$. Prove that if $u, v \in \operatorname {Lip}_{\alpha }(\left[0,1\right])$, then $\alpha \leqslant \frac{1}{2}$. (Thus, the smoothness of the coordinate functions of the Peano curve constructed in the previous problem is the best possible).

Let $f$ be a continuous real function on $\mathbb {R}^{1}$ with the following properties: $0 \leq f(t) \leq 1, f(t+2) = f(t)$ for every $t$, and

Put $\Phi (t) = (x(t), y(t))$, where

Prove that $\Phi$ is continuous and that $\Phi$ maps $I = [0,1]$ onto the unit square $I^{2} \subset \mathbb {R}^{2}$. If fact, show that $\Phi$ maps the Cantor set onto $I^{2}$.

Hint: Each $\left(x_{0}, y_{0}\right) \in I^{2}$ has the form

where each $a_{i}$ is 0 or 1. If

show that $f\left(3^{k} t_{0}\right) = a_{k}$, and hence that $x\left(t_{0}\right) = x_{0}, y\left(t_{0}\right) = y_{0}$. (This simple example of a so-called "space-filling curve" is due to I. J. Schoenberg, Bull. A.M.S., vol. 44, 1938, pp. 519).