f[t_] /; 0≤t≤1/3 := 0;
f[t_] /; 1/3<t<2/3 := 3t-1;
f[t_] /; 2/3≤t≤1 := 1;
f[t_] /; 1<t<2 := f[2-t];
f[t_] /; t≥2 := f[2FractionalPart[t/2]];

Plot[f[t],{t,0,9}]

x[n_,t_] := Sum[1/2^k f[3^(2k-2)t], {k,1,n}];
y[n_, t_] := Sum[1/2^k f[3^(2k-1)t], {k,1,n}];
Peano[n_] := ParametricPlot[{x[n,t], y[n,t]}, {t,0,1}]

Table[Peano[n], {n,4}]