Let $f(x,y)$ be a smooth function. It is given that for every $x$ there exists at least one $y$ such that $f(x,y)=0$. Is this possible to select one such $y$ for every $x$, such that the $y$'s are a smooth function of $x$?
I.e., is there a smooth function $Y(x)$, such that for every $x$: $f(x, Y(x))=0$?
Intuitively, if $f(x,y)$ describes the height of a certain landscape at coordinates $(x,y)$, then the zeros are the sea-level locations, and the question is: can one walk from west to east in a smooth path which is always on the sea-level?