Let $f\colon \mathbb R^2 \to \mathbb R$ be a real-analytic function of two-variables. Further suppose that $f(x,y)=0$ has at least one solution for all $x$ in some neighborhood $B_r = (x_0-r,x_0+r)$. Call this largest solution $\gamma(x)$. I believe that $\gamma \colon B_r \to \mathbb R$ defines a curve.
Is $\gamma$ continuous at $x_0$? How about smooth?
