Calculating the derivative, it's onto as long as $c$ isn't 1 or -1.
But when c = 1 or -1, is it still a smooth curve?
EDIT: I wrote c = 0 for some stupid reason
2026-04-04 13:07:40.1775308060
For what values of $c$ is $\sin (x+y) = c$ a smooth curve? Asked
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It is a bunch of parallel lines, all of slope $-1$. When $c$ reaches $\pm1$, the lines collide in pairs; as $c$ passes $-1$ or $1$, they disappear.
So they are always smooth, if they exist.