Limits and Discontinuities in Surfaces

Discontinuous Functions

Plot a surface that is discontinuous at (0,0) due to a singularity where the graph blows up to infinity.  Note that the surface has a catastrophic discontinuity at (0,0) and that the function does not have a limit as (x,y) tends to (0,0). But this fails only on certain paths tending to (0,0).

f[x_,y_]:=x*y^2/(x^2 + y^6);

Plot3D[f[x,y],{x,-1,1},{y,-1,1},AxesLabel->{"x","y","f(x,y)"}, PlotLabel->"The Surface xy^2/(x^2 + y^6)", ColorFunction->Function[{x,y,z},Hue[z]]]

Plot the surface along the lines y = x and y = (x^2) by graphing f(x,x) and f(x,(x^2)) resepectively in two dimensions.  Note that no singularity exists along these paths, though the behavior certainly differs :

Plot[f[x,x],{x,-1,1},AxesLabel->{"x","f[x,x]"}, PlotStyle->Thick, PlotLabel->"Perspective graph of f[x,y] along the path y = x approaching the origin"]

Plot[f[x,x^2],{x,-1,1},AxesLabel->{"x","f[x,x]"}, PlotStyle->Thick, PlotLabel->"Perspective graph of f[x,y] along the path y = x^2 approaching the origin"]

Plot the surface along the lines y = (x^(1/3)) by graphing f(x,(x^(1/3))) in two dimensions.  Note that along this path, the graph blows up to +∞.  This is verified by using the Limit function in Mathematica :

Plot[f[x,x^(1/3)],{x,.00001,.001},PlotRange->{0,30},PlotStyle->Thick, AxesLabel->{"x","f[x,x]"}, PlotLabel->"Perspective graph of f[x,y] along the path y = x^(1/3) approaching the origin"]

Limit[f[x,x^(1/3)],x->0]

Use Mathematica multivariable limit functionality (if it works) :

Limit[f[x,y],{x,y}->{0,0}]

Plot a surface that is discontinuous at (0,0) due to differing limits along different paths, but NOT due to the graph blowing up to infinity :

g[x_,y_]:=x*y/(x^2+y^2);

Plot3D[g[x,y],{x,-1,1},{y,-1,1},AxesLabel->{"x","y","g(x,y)"}, PlotLabel->"The Surface xy/(x^2 + y^2)", ColorFunction->Function[{x,y,z},Hue[z]]]

Along the path y = x, the graph has constant value g(x,x) = 1/2 :

Plot[g[x,x],{x,-1,1},AxesLabel->{"x","g[x,x]"}, PlotStyle->Thick, PlotLabel->"Perspective graph of g[x,y] along the path y = x approaching the origin", PlotRange->{0,1}]

Along the path y = 4x, the graph has constant value f(x,x) = 4/17 :

Plot[g[x,4x],{x,-1,1},AxesLabel->{"x","g[x,x]"}, AxesOrigin->{0,0},PlotRange->{0,1},PlotStyle->Thick, PlotLabel->"Perspective graph of g[x,y] along \nthe path y = (x^2) approaching the origin"]

Along the path y = (x^2), the graph has the limit of 0 as x tends to 0 :

Plot[g[x,x^2],{x,-1,1},AxesLabel->{"x","g[x,x]"}, PlotStyle->Thick, PlotLabel->"Perspective graph of g[x,y] along \nthe path y = (x^2) approaching the origin"]

Use Mathematica multivariable limit functionality (if it works) :

Limit[g[x,y],{x,y}->{0,0}]

Continuous Functions

A function that is continuous at (0,0):

h[x_,y_]:=x*y/Sqrt[x^2+y^2];

Plot corresponding surface.  Note that although this surface has a “wrinkle” at (0,0), the function is continuous at (0,0).

Plot3D[h[x,y],{x,-1,1},{y,-1,1},AxesLabel->{"x","y","h(x,y)"}, PlotLabel->"The Surface xy/(x^2 + y^2)^(1/2)", ColorFunction->"TemperatureMap"]

Plot the above surface along the line y = x by graphing h(x,x) in two dimensions.

Plot[h[x,x],{x,-1,1},AxesLabel->{"x","h[x,x]"}, PlotStyle->Thick, PlotLabel->"Perspective graph of h(x,y) along the path y = x approaching the origin"]

Plot the above surface along the line y = (x^2) by graphing f (x, (x^2)) in two dimensions.  Note the smooth curve when approaching (0,0) along this path :

Plot[h[x,x^2],{x,-1,1},AxesLabel->{"x","h[x,x]"}, PlotStyle->Thick, PlotLabel->"Perspective graph of h[x,y] along the path y = (x^2) approaching the origin"]

Use Mathematica multivariable limit functionality (if it works) :

Limit[h[x,y],{x,y}->{0,0}]