Exercise 9.20

If the real-valued function $f(x,y)$ is smooth and nonconstant in a region of $2$, then the solution of $f(x,y)=0$ is locally a smooth curve. If $D_{1}f(x_0,y_0)\ne 0$ at a point of the curve, then the curve doesn’t have a vertical tangent at $(x_0,y_0)$, and so it will be the graph of a function $y=g(x)$ near $x=x_0$, so that $f(x,g(x))=0$. Similarly, if $D_{2}f(x_0,y_0)\ne 0$, then the curve doesn’t have a horizontal tangent at $(x_0,y_0)$, and so it will be the graph of a function $x=h(y)$ near $y=y_0$, so that $f(h(y),y)=0$.