real analysis – Smoothness of a unique level set of a smooth function

Suppose that $f:mathbb{R}^2to mathbb{R}$ is smooth and, for all $xin mathbb{R}$, there exists a unique $y(x)$ such that $f(x,y(x))=0$. In other words, the graph of a continuous function $y=y(x)$ is the $0$ level set of $f$. Is it true that $y(x)$ is differentiable $x$-almost everywhere?