File:Uniform continuity animation.gif

English: animation showing a discontinuous function as a pointwise limit of continuous functions

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]} ]} ];