Consider the worksheet below containing a function that I came across while studying Apostol's Calculus.
At the origin, this function has a defined directional derivative in all directions. It is not, however, continuous at the origin. We can see this by consider all points on the parabola x=y^2 except for the origin. The function takes on the value 1/2 on all such points but has value 0 at the origin and is thus discontinuous there.
My question is about a 3d plot of this function.
The plot seems a bit inaccurate because the ridge at the top extends all the way to the origin.
If I hadn't done the calculations to know this, this plot would not give me this information.
Is there a way to avoid this problem? Ie, to get more accuracy at points such as the origin here?