Check if the parametrization of the curve $\gamma :\mathbb{R}\rightarrow\mathbb{R}^{2}$ defined by $$\gamma(t):=(\frac{t}{1+t^{4}},\frac{t}{1+t^{2}})$$ is an homeomorphism on $Im(\gamma)$
This map is injective and trivally surjective. It is also continuous but I don't know how to see if $\gamma^{-1}$ is continuos or not. I can't find the inverse function and it's not easy to see if it's open or closed. Can someone help me? Thanks before!
It is not an homeo : as $t\to \infty$ $\gamma(t)\to (0,0)=\gamma(0)$. In particular,one can find points are arbitrarly close to $\gamma(0)$ for instance $\gamma (n)= x_n$ but whose image by $\gamma^{-1}$ (in our example $\gamma^{-1}(x_n)=n$) are very far from the origin.