I need to prove that $\sin(xy)+\cos(x+y)=x^2+y$ has infinitely many solutions in x,y and the set of all solutions is a closed set in $\mathbb{R^2}$
Proving is closed seems fairly easy. $f: \mathbb{R^2}\rightarrow\mathbb{R} $ $f(x,y)=\sin(xy)+\cos(x+y)-x^2-y=0$.
$A=\{0\}$ is closed so $f^{-1}(A)$ is also closed by continuity.
Now, I have no idea how to prove that there are infinitely many solutions. Any ideas?