# 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?