File:Uniform continuity animation.gif

Summary

animation showing a discontinuous function as a pointwise limit of <a href="//commons.wikimedia.org/wiki/Continuous_function" title="Continuous function">continuous functions</a>

mathematica source code:

f[n_, x_] := Which[x <= 0,

  0, 0 < x < 1/n, n x, 1/n ≤ x, 1]; Table[Plot[f[n^2, x - 
   1] + 1, {x, -1, 3}, 
     PlotRange -> {{-1, 3}, {0, 2.2}}, PlotStyle -> {Thickness[
         0.02]}, Ticks -> False], {n, 1, 10}]

Show[{Plot[f[100, x - 1] + 1, {

     x, -1, 0.9}, 
         PlotRange -> {{-1, 3}, {0, 2.2}}, PlotStyle -> {Thickness[0.02]}, \

Ticks -> False],

     Plot[f[100, x - 1] + 1, {x, 1.1, 3}, PlotRange -> {{-1, 3}, {0,
2.2}}, PlotStyle -> {Thickness[0.02]}, Ticks -> False],
     Graphics[Circle[{1, 2}, 0.3]]}
   ];

Show[{Plot[f[100, x - 1] + 1, {x, -1, 0.9}, PlotRange -> {{-1, 3}, {0, 2.2}},

         PlotStyle -> {Thickness[0.02]}, Ticks -> False],
     Plot[f[100, x - 1] + 1, {x, 1.1,
      3}, PlotRange -> {{-1, 3}, {0, 2.2}}, PlotStyle -> {Thickness[0.02]}, 
       Ticks -> False],
     Graphics[{
         Disk[{1, 1}, 0.07],
         Circle[{1, 2}, 0.07]}
       ]}
   ];

Licensing

Lua error in package.lua at line 80: module 'strict' not found.