Discontinuous Surface with Parabolic Contours

Discontinuous Surface with
Parabolic Contours

12.3 Limits and continuity

Note: Section numbers refers to Calculus, Early Transcendentals (2nd edition) by Briggs, Cochran, and Gillett.

This function is discontinuous at (x,y)=(0,0). The limit of f(x,y) along every straight line through the origin is 0 (as shown for the three lower yellow curves). However, the function has a different limit along every parabola through the origin - as shown by the upper yellow curve. In fact, the parabolas through the origin are exactly the level sets of the surface.