Is $\{ (x,y) \in \mathbb{R}^2 : \sin \left(\frac{x^3}{x^4+y^2}\right) >0 \}$ an open subset of $\mathbb{R}^2$ with the Euclidean metric?

Question

Is $\{ (x,y) \in \mathbb{R}^2 : \sin \left(\frac{x^3}{x^4+y^2}\right) >0 \}$ an open subset of $\mathbb{R}^2$ with the Euclidean metric?

I think I have to use the theorem about if a function is continuous, its preimages are open, but I am not sure how to deal with the point $(0,0)$ since we don’t have continuity there...

Answer 1

If we define $f(x,y) = \sin(\frac{x^3}{x^4 + y^2})$ where $f: \mathbb{R}\setminus \{(0,0)\} \to \mathbb{R}$ ,then $f$ is continuous on the domain of $f$.

I think your set $A:= \{ (x,y) \in \mathbb{R}^2 : \sin \left(\frac{x^3}{x^4+y^2}\right) >0 \}$ cannot contain $(0,0)$ because we cannot substitute $(0,0)$ in the formula. (And also, $f$ does not have a continuous extension to all of $\mathbb{R}^2$, so there is no "way out" that way.)

So $A = f^{-1}[(0,\infty)]$ which is open in the domain of $f$ by continuity of $f$, and as the domain is open in $\mathbb{R}^2$ (singletons are closed), $A$ is also open in $\mathbb{R}^2$.