In multivariable calculus, we learn to find local extrema by identifying the critical points, and deciding (using the second derivative test, or otherwise) the type of the point - a local max, a local min, or a saddle point.
What if we get a continuum of critical points ? say the gradient of $f(x,y)$ vanishes along a curve in the $xy$ plane. Is this situation possible ? what can we say about classifying these points into min/max/saddle ?