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Jan 9 at 17:43 comment added fedja @IosifPinelis "Can you draw such a contour plot?" Just curve the crossing straight lines. The picture is topological, not metric, so you can make any curve escaping to $\infty$ a straight line and any 3 points lie on such curve.
Jan 9 at 12:47 comment added Iosif Pinelis I agree that this is rigorous, and I did think about this possibility. However, the realization of this approach seems rather laborious, especially the gluing should be done rather carefully, and the eventual product will probably be a piecewise-defined function with complicated expressions, especially after the gluing. My efforts were directed to finding a very simple expression for a function in question. Of course, I am being apologetic here. :-)
Jan 9 at 12:20 history answered fedja CC BY-SA 4.0