Straight and Parabolic Limits

Here z = x^2 y / (x^4 + y^2), the classic example of a discontinuous function whose straight-line limits into the origin are all 0, but which has different limiting values on each parabola. The parabolic contours are evident in the image. We first generated the graph using closed forms for the parabolas and ellipses that form the contours and flows, but it is more general to use numerical methods, and that is how the present image here was done. The contours are color-coded from cyan to red to aid the visualization of height. As with many of our examples, a plot using standard rectangular coordinates is totally unsatisfactory.