Please, construct a continuous piecewise linear function g.
Select the number of pieces:

Choose the intermediate points (they must strictly increase)
and the values of g at the intermediate points:

Graph of the generating function g:

Graph of the quantile function Q:

Show the eigenvalues λ_{n,1},…λ_{n,n}
of the Toeplitz matrix of order n:

The eigenvalues are computed with
Jacobi algorithm.
For each j=1,…,n, the jst eigenvalue λ_{n,j}
is shown as the point with abscissa j/(n+1) and ordinate λ_{n,j}.
We proved that if Q is continuous,
then the difference λ_{n,j}−Q(j/(n+1)) tends to 0 uniformly in j as n tends to ∞.