In order to complete a proof regarding the existence of a plane-filling curve (that is, a continuous and surjective mapping from $\Bbb R$ to $\Bbb R^2$) I came across the following problem:
“Suppose there exists a space-filling curve $\phi : [0,1] \rightarrow [0,1]^2$ then, from this only assumption can we prove that there exists a space-filling curve $\phi’$ such that $\phi’(1)=(0,0)$ and $\phi’(0)=(0,0)$?”
I am sorry if I do not seem well-informed on the topic, but it is my first time dealing with it.
As always any help in proving or disproving this statement is highly appreciated!
Edit I added the condition $\phi’(0)=(0,0)$ which I originally forgot while writing the question.
Well, you can take $\phi'(t)=\begin{cases} \phi(2t) & \text{if } 0\le t\le \frac12\\ (2-2t) \phi(1) & \text{if } \frac12 <t\le 1 \end{cases}$