Please edit your text using "[ ]" in place of "< >" for the HTML tags. It is extremely difficult to read as it is.
Suppose that z = f(x,y) is a continuous surface over some closed, rectangular domain in the xy-plane. Let Pf be the projection of f onto the xz-plane, i.e. (x,0,z) is a point of Pf if and only if (x,y,z) is a point of f. Is the boundary of Pf the union of two continuous functions? The "upper" boundary of Pf;is made up of points which are the maxima of the cross-sections of the surface f, perpendicular to the x-axis. The "lower" boundary of Pf is made up of points which are the minima of the cross-sections of the surface f, perpendicular to the x-axis. Since f is continuous and the domains of the above cross-sections are closed intervals, therefore these extrema are well defined. But, do the union of these extrema form continuous curves? Intuition tells me that the answer is yes. Intuition also tells me that, if we replace continuity by differentiability then the answer is probably no. Has anybody seen this problem before? Is this problem trivial? Does the solution depend on uniform continuity?
I would like to do as you asked. Had I known that the posted version of my question and the version, as I wrote it would be different, I would not have used any subscripts or quotes.
I would like to edit what I wrote, but I am not sure how I can do this.
Thank you for your consideration and time.