This notebook shows Euler's Broken Line for the (easiest) example:     x'=x, x(0)=1.

α = 1/2

phi[t_]=Which[0≤t≤1/2,1+t,1/2≤t≤1,phi[1/2]+Integrate[1+s,{s,1/2,t}]]

Plot[phi[t],{t,0,1}]

phi[1]//N

α = 1/n

Clear[p]
n=5

p[1]=Function[t,1+t]

p[k_]:=p[k]=Function[t,p[k-1][(k-1)/n]+Integrate[p[k-1][s],{s,(k-1)/n,t}]//Expand]

phi[t_]=Piecewise[Table[{p[k][t],(k-1)/n≤t≤k/n},{k,1,n,1}]]

Plot[{Exp[t],phi[t]},{t,0,1},PlotStyle->{Dotted,Thick},ImageSize->800]
Print["At t=1, Euler's Broken Line for n= ",n," evaluates to ",phi[1.]]
Print["e = ",Exp[1.]]