# A curve filling a square – Lebesgue example

## Introduction

We aim at defining a continuous function $$\varphi : [0,1] \rightarrow [0,1]^2$$. At first sight this looks quite strange.

Indeed, $$\varphi$$ cannot be a bijection. If $$\varphi$$ would be bijective, it would also be an homeomorphism as a continuous bijective function from a compact space to a Haussdorff space is an homeomorphism. But an homeomorphism preserves connectedness and $$[0,1] \setminus \{1/2\}$$ is not connected while $$[0,1]^2 \setminus \{\varphi(1/2)\}$$ is.

Nor can $$\varphi$$ be piecewise continuously differentiable as the Lebesgue measure of $$\varphi([0,1])$$ would be equal to $$0$$.

$$\varphi$$ is defined in two steps using the Cantor space $$K$$. Continue reading A curve filling a square – Lebesgue example