How to mark visible points on a Plot3D?

Question

I'd like to illustrate a two-variable function discontinuous at zero. The function is $f(x,y)=xy/(x^2+y^2)$ unless $x=y=0$, in which case $f(0,0)=0$. I thought that I could draw a plot and then mark two sequences of points $(x_n,y_n)$ and, say, $(x^\prime_n,y^\prime_n)$ converging to $(0,0)$ such that $f(x_n,y_n)$ and $f(x^\prime_n,y^\prime_n)$ converge to different limits. I did something like this:

f[x_, y_] := If[x == y == 0, 0, x*y/(x^2 + y^2)]
p := Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}]
Show[p, ListPointPlot3D[
  Thread[{Table[1/n, {n, 5}], Table[1/n, {n, 5}], 
    Table[f[1/n, 1/n], {n, 5}]}]]]

but the dots are barely visible; I'd like to have them bigger and/or blacker. How to achieve this?

Answer 1

An alternative to Verbeia's answer is to use Sphere[] primitives to depict the points, as in the following:

f[x_, y_] := If[x == y == 0, 0, x y/(x^2 + y^2)];

Show[Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}], 
     Graphics3D[{Darker[Blue], Sphere[Table[{1/n, 1/n, f[1/n, 1/n]}, {n, 5}], 1/15]}], 
     PlotRange -> All]

---

This is a bit beyond your actual question, but let me just note that the surface you're considering is one of those surfaces that look better in cylindrical coordinates than in rectangular coordinates. Witness the following:

Show[ParametricPlot3D[{r Cos[θ], r Sin[θ], Cos[θ] Sin[θ]}, {r, 0, Sqrt[2]}, {θ, -π, π}], 
     Graphics3D[{Darker[Blue], Sphere[Table[{1/n, 1/n, f[1/n, 1/n]}, {n, 5}], 1/15]}], 
     PlotRange -> All]

Answer 2

Nasser's suggestion in comments is correct. You need to provide a non-default style to the points in the ListPointPlot3D. I would also suggest giving p a style that allows one to see through the surface to the points partly on the other side of the surface, using Opacity. And incidentally, there's no particular reason to use SetDelayed (:=) rather than Set (=) when defining p. You can suppress output with a semicolon. If you use SetDelayed, you are recalculating the graphic when you evaluate the Show expression, which is a waste of processor cycles.

f[x_, y_] := If[x == y == 0, 0, x*y/(x^2 + y^2)]

p = Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}, 
   PlotStyle -> Opacity[0.5]] ;

Show[p, ListPointPlot3D[
  Thread[{Table[1/n, {n, 5}], Table[1/n, {n, 5}], 
    Table[f[1/n, 1/n], {n, 5}]}], PlotStyle -> PointSize[.05]]]

As you can see, options are placed at the end of a function list. This tutorial is about writing your own functions to take options and optional arguments, but it helps explain how options work. The Mathematica documentation for each function (including ListPointPlot3D) goes through the options it can take and provides examples.