Code:

Show[Table[{Plot[t, {t, 2 j \[Pi], (2 j + 1) \[Pi]},
PlotStyle -> Red],
Plot[t - 2 \[Pi], {t, 2 j \[Pi], (2 j + 1) \[Pi]},
PlotStyle -> Blue],
Plot[0, {t, 2 j \[Pi], (2 j + 1) \[Pi]},
PlotStyle -> {Black}]}, {j, 0, 3}],
Ticks -> {Table[j \[Pi], {j, 0, 7}], Table[j \[Pi], {j, -2, 7}]},
PlotRange -> {{0, 7 \[Pi]}, {-2 \[Pi], 7 \[Pi]}}, AspectRatio -> 7/9,
Epilog ->
Inset[Show[
Table[Plot[0, {x, j \[Pi], j \[Pi] + \[Pi]},
PlotRange -> {{0, 7 \[Pi]}, {-2 \[Pi], 7 \[Pi]}}, Axes -> False,
PlotStyle -> {{Thickness[.006], White, EdgeForm[]}},
AspectRatio -> 7/9], {j, 1, 7, 2}]], {0, 0}, {0, 0}, {7 \[Pi],
Automatic}]
]

I.e., draw white lines over the parts of the axis you don't want. If you use it to produce a vector graphic to include elsewhere, this might be a problem, though.