A Function That Is Continuous But Not Differentiable

Define Function f(x,y) = (x^3)/((x^2)+(y^2)) :

f[x_,y_]= x^3/(x^2+y^2);

Plot Surface Around (0,0).  Note the continuous “wrinkle” near the origin.  There is no hole or discontinuity in the graph. But there is also no tangent plane at (0,0).

SurfPlot = Plot3D[f[x,y],{x,-2,2},{y,-2,2},ColorFunction->Function[{x,y,z},Hue[z]], AxesLabel->{"x","y","z"}, PlotLabel->"f(x,y) = x^3/(x^2+y^2) - Continuous at (0,0), but not Differentiable"]

Check Continuity Of Partial Derivatives.  At least one must be discontinuous.  Otherwise, by the theorem in class, if they were both continuous, then f would be differentiable.

Compute ∂f/∂x using the “D” operator in Mathematica :

Simplify[D[f[x,y],x]]

Plot ∂f/∂x.  Notice the discontinuity at (0,0) :

SurfPlot = Plot3D[(((x^4)+3 (x^2) (y^2))/((((x^2)+(y^2)))^2)),{x,-2,2},{y,-2,2},ColorFunction->Function[{x,y,z},Hue[z]], AxesLabel->{"x","y","z"}, PlotLabel->"∂f/∂x - not continuous at (0,0)"]

Compute ∂f/∂y using the “D” operator in Mathematica :

Simplify[D[f[x,y],y]]

Plot ∂f/∂x.  Notice the discontinuity at (0,0) :

SurfPlot = Plot3D[-((2 (x^3) y)/((((x^2)+(y^2)))^2)),{x,-2,2},{y,-2,2},ColorFunction->Function[{x,y,z},Hue[z]], AxesLabel->{"x","y","z"}, PlotLabel->"∂f/∂y - not continuous at (0,0)"]