Is the set $ \{x \in \mathbb{R^2} | x_1 + x_2 < x_2^2 \} $ open or closed?
This question has been confusing me for a very long time. I have tried solving it using coordinate geometry and trying to find the shortest distance between any arbitrary point x and the parabola created by $ x_1 + x_2 = {x_2}^2 $ but it just results in a huge mess. It so extremely obvious to me that the set is open when I draw a picture but I am not able to formally prove it.
Any help would be extremely appreciated.
Consider the continuous function $\varphi(x_{1},x_{2})=x_{1}+x_{2}-x_{2}^{2}$, the preimage of $(-\infty,0)$.