Show that the curve $t\rightarrow (t,t^2,t^3)$ embeds $\mathbb{R}$ into $\mathbb{R^3}$. Find two independent functions that globally define the image. Are your functions independent on all of $\mathbb{R^3}$ or just an open neighborhood of the image?
My Try:
To show that it is an embedding I must show it is continuous and a homeomorphism onto its image. I think it is obvious. But I do not have any clue on how to approach the rest. Can anybody please help me?
"I think it is obvious": the continuous part maybe, but the fact that it is a homeomorphism with its image is arguably not self-evident. Luckily, the inverse in this case is rather simple (what is it?), so it is easy to show that it is a homeomorphism with its image.
Probably the "embedding" part also wants you to show that it is an immersion since you tag in differential topology, so you should also check that.
Now, about finding two independent functions: you want $x^2=y$ and $x^3=z$ in that curve, correct? Use that to define the functions.