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]}
]}
];