DescriptionHomogeneousDiscontinuousFunction.gif	English: An example that shows that a homogeneous function does not have to be continuous. This is the function f defined by f(x,y)=x if xy>0 or f(x,y)=0 otherwise. It is homogeneous of order 1, i.e. f(a*x,a*y)=a*f(x,y). It is discontinuous at y=0, x=/=0.
Date	16 February 2011
Source	Own work  This diagram was created with Mathematica.
Author	Sbyrnes321

(* Source code in Mathematica 6.0 by Steve Byrnes, 2011. *) 
eps = .001;
a = Show[Plot3D[x, {x, 0, 1}, {y, eps, 1}, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}],
  Plot3D[0, {x, -1, 0}, {y, eps, 1}],
  Plot3D[0, {x, 0, 1}, {y, -1, -eps}],
  Plot3D[x, {x, -1, 0}, {y, -1, -eps}],
  Plot3D[x (1 - y/eps)/2, {x, -1, 0}, {y, -eps, eps}],
  Plot3D[x (1 - y/eps)/2, {x, 0, 1}, {y, -eps, eps}]]
Export["pic.gif", a]

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