TITLE: How to show that $\{(x, y) \in \mathbb{R}^2\mid\cos(x^2) + x^3 - 4 7y > e^x - y^2\}$ is open or closed?
QUESTION [2 upvotes]: Show that the set of points $(x, y) \in \mathbb{R}^2$  such that $\cos(x^2) + x^3  - 4 7y > e^x - y^2$  is an open subset of $\mathbb{R}^2$.
I was reading the fact that if $(X,d)$ and $(Y,d)$ are metric spaces and $f:(X,d) \to \rightarrow (Y,d)$ is continuous and if $v$ be a open set (or closed set) in $Y$ then $f^{-1}(v)$ is open (closed) in $X$. I don't really under stand how to do the applications for this lemma. So far I have proved using this theorem that the identity function and the constant function are continuous. I want help to understand this problem and for similar problems I want to know what I am looking for here.what are the steps to show that a given set is closed or open using this lemma.

REPLY [1 votes]: The other answers show how to do it for this specific example, but it’s helpful to see it as a general principle: A strict inequality between continuous functions always defines an open set.
Precisely, given continuous functions $l, r : X \to \mathbb{R}$ on any metric space $X$ (more generally, any topological space), the set $\{ x \in X\ |\ l(x) < r(x) \}$ is open.  This is because the function $f : X \to \mathbb{R}$ given by $f(x) := r(x) - l(x)$ is continuous, and $\{ x \in X\ |\ l(x) < r(x) \} = \{ x \in X\ |\ f(x) > 0 \} = f^{-1}(0,\infty)$.
Exercise: with a similar argument, show that a non-strict inequality between continuous functions always defines a closed set, i.e. that any set of the form $\{ x \in X\ |\ l(x) \leq r(x) \}$ (for continuous functions $l, r : X \to \mathbb{R}$) is closed.
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