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?
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 $\Bbb R$.