Discontinuity

Author

Eric W. Weisstein
August 25, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/Discontinuity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Discontinuity.html.

©2014 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Discontinuity types from
Christopher Stover
August 2014

Example

GraphicsRow[{
Plot[Piecewise[{{Sin[x],x<2}},Sin[x]-1/4],{x,0,6},LabelStyle->{Black,FontFamily->"Times"},Epilog->{Red,Dashing[{.04}],Line[{{2,-2},{2,2}}]}],
Plot3D[Im[Log[x+I y]],{x,-5,5},{y,-5,5},PlotPoints->30,
PlotLabel->Im[ln[z]],
AxesLabel->{x,y,""},LabelStyle->{Black,FontFamily->"Times"}]
},ImageSize->400
]

Types

Infinite Discontinuity

Plot[Piecewise[{{1.1,x==0}},Sin[1/x]],{x,-1,1},LabelStyle->{Black,FontFamily->"Times"},PlotStyle->Red,Epilog->{Red,PointSize[0.0125],Point[{0,1.1}]}]

Infinite Line discontinuity

GraphicsRow[{
Plot3D[1/(x-y),{x,-10,10},{y,-10,10},Exclusions->{x==y},LabelStyle->{Black,FontFamily->"Times"},ClippingStyle->None,PlotLabel->h[x,y],AxesLabel->{x,y}],
Plot3D[1/((x-y)^2),{x,-10,10},{y,-10,10},Exclusions->{x==y},LabelStyle->{Black,FontFamily->"Times"},ClippingStyle->None,PlotLabel->k[x,y],AxesLabel->{x,y}]
},
ImageSize->400]//Quiet

Infinite Point discontinuity

GraphicsRow[{
Plot3D[1/(x^2+y^2),{x,-10,10},{y,-10,10},LabelStyle->{Black,FontFamily->"Times"},ClippingStyle->None,PlotLabel->f[x,y],AxesLabel->{x,y}],
Plot3D[Sin[1/Sqrt[x^2+y^2]],{x,-10,10},{y,-10,10},LabelStyle->{Black,FontFamily->"Times"},ClippingStyle->None,PlotLabel->g[x,y],AxesLabel->{x,y}]
},ImageSize->400]

Monotone Line Jump discontinuity

f[x_,y_]=Piecewise[{{{x,x},x+y>1},{{x-5,x-5},x+y≤1}}]

Plot3D[f[x,y],{x,-10,10},{y,-10,10},LabelStyle->{Black,FontFamily->"Times"},PlotLabel->t[x,y],AxesLabel->{x,y}]

Pinch Discontinuity

Plot3D[(2*x*y)/(x^2+y^2),{x,-10,10},{y,-10,10},LabelStyle->{Black,FontFamily->"Times"},PlotLabel->p[x,y],AxesLabel->{x,y}]

Removable Discontinuity

Plot[(x^2-1)/(x-1),{x,-3,3},PlotStyle->Red,PlotRange->3,Epilog->{Red,Disk[{1,2.5},.05],White,EdgeForm[Red],Disk[{1,2},.05]},AspectRatio->Automatic]