The map $(x,y)\mapsto(x,y,x^2+y^2)$ is a continuous surjection from the <path-connected> plane onto this surface. It is a <homeomorphism> because coordinate projection is its continuous inverse. Thus \b[the paraboloid is <path-connected> and <connected>]. Explicitly, lifting any straight planar segment by this map produces a path on the surface.

Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 2 / 4F / iii / Solution

 Solution (source code)