Stetigkeit von Funktionen

Beispiele für Unstetigkeiten:

f (x, y)=((x + y)/((x^2) + (y^2)))

Show[
      GraphicsArray[{{Plot3D[(x + y)/(x^2 + y^2), {x, -1, 
                1}, {y, -1, 1}, PlotPoints -> 100, 
              PlotRange -> {-20, 20}, 
              DisplayFunction -> 
                Identity]}, {ContourPlot[(x + y)/(x^2 + 
                    y^2), {x, -1, 1}, {y, -1, 1}, 
              PlotPoints -> 100, Contours -> 25, 
              DisplayFunction -> Identity, 
              ColorFunction -> (Hue[#/4] &)]}}], 
      ImageSize -> 800]

f (x, y)=(((x^2) - 2  (y^2))/((x^2) + (y^2)))

Show[
      GraphicsArray[{{Plot3D[(x^2 - 2  y^2)/(x^2 + y^2), {x, -1, 
                1}, {y, -1, 1}, PlotPoints -> 100, 
              PlotRange -> {-2, 2}, 
              DisplayFunction -> 
                Identity]}, {ContourPlot[(x^2 - y^2)/(x^2 + 
                    y^2), {x, -1, 1}, {y, -1, 1}, 
              PlotPoints -> 100, Contours -> 25, 
              DisplayFunction -> Identity, 
              ColorFunction -> (Hue[#/4] &)]}}], 
      ImageSize -> 800]

f (x, y)=(((x^2) y)/((x^4) + (y^2)))

Show[
      GraphicsArray[{{Plot3D[(x^2  y)/(x^4 + y^2), {x, -1, 
                1}, {y, -1, 1}, PlotPoints -> 100, 
              DisplayFunction -> 
                Identity]}, {ContourPlot[(x^2  y)/(x^4 + 
                    y^2), {x, -1, 1}, {y, -1, 1}, 
              PlotPoints -> 100, Contours -> 25, 
              DisplayFunction -> Identity, 
              ColorFunction -> (Hue[#/4] &)]}}], 
      ImageSize -> 800]

Beispiel für Stetigkeit:

f (x, y)=(((x^2) y)/((x^2) + (y^2)))

Show[
        GraphicsArray[{{Plot3D[(x^2  y)/(x^2 + y^2), {x, -1, 
                  1}, {y, -1, 1}, PlotPoints -> 100, 
                DisplayFunction -> 
                  Identity]}, {ContourPlot[(x^2  y)/(x^2 + 
                      y^2), {x, -1, 1}, {y, -1, 1}, 
                PlotPoints -> 100, Contours -> 25, 
                DisplayFunction -> Identity, 
                ColorFunction -> (Hue[#/4] &)]}}], 
        ImageSize -> 800]