How to prove that a curve is an injective regular parameterization?

Question

I have the above assignment: Prove that the curve $σ:\Bbb R → \Bbb R^2$ given by $$σ(t)=\left(\frac{t}{1+t^4}, \frac{t}{1+t^2}\right),$$ is an injective regular parameterization, but not a homeomorphism with its image.

I have found the derivative and prove that it does not equal to zero so it is a regular parameterization but I don't know what to do next.

Can someone help?

Answer 1

Once one get the plot, one sees that the trouble is at the $\sigma(0)$ position.

There, is impossible to find an open set $U$ in $\Bbb R^2$ such that ${\rm im}\sigma\cap U$ is an open arc containing $\sigma(0)$, i.e. there $\sigma$ isn't locally homeomorphic to $\Bbb R$.

Answer 2

I don't know if this is any good to you now but it may help others.

The parametrization is injective as $ \forall t \in $ $\Bbb R$ as $\exists$ a unique x(t) , y(t) in $\Bbb R^{2}$ where $ x(t) = \frac{t}{1+t^{4}} $ , $ y(t) = \frac{t}{1+t^{2}} $ this can be seen by plotting the function.

This question can be found in Curves and Surfaces by Abate and Tovena problem 1.1