For a function $f: \mathbb{R}^n \to \mathbb{R}$, I was wondering
Is $\{x \in \mathbb{R}^n: f(x) < 0\}$ open?
If not, what are some sufficient and/or necessary conditions for it to be open?
Is $\{x \in \mathbb{R}^n: f(x) = 0\}$ closed?
If not, what are some sufficient and/or necessary conditions for it to be closed?
Is $\{x \in \mathbb{R}^n: f(x) \leq 0\}$ closed?
If not, what are some sufficient and/or necessary conditions for it to be closed?
- I feel $\{(x,y) \in \mathbb{R}^2: -x+\sqrt{y} < c \}$ for some $c \in \mathbb{R}$ is open. To prove it, I want to show that for every point in it, it has an open ball centered at it and the ball is contained inside the subset. This is however not obvious for me to show.
Thanks and regards!