I the element $h_{f,i}$ is given by: $$\begin{aligned}
h_{f,i}&=&(y_p(i)-y_p(i-1)) \left(\frac{x-x_p(i-1)}
{x_p(i)-x_p(i-1)}\right)^{\alpha(i)} + y_p(i-1)
\nonumber \\
~~~x&\in&[x_p(i-1),x_p(i))~~;~~~i=1,2,..,N
\label{hfexp}\end{aligned}$$ where $y_p(i-1),y_p(i)$ are the values of $h_f$ at the points $x_p(i-1),x_p(i)$ respectively, while the power $\alpha(i)$ determines the local behaviour of $h_f$ around $x_p(i-1)$. Due to the constraints (\[hfprei\],\[hfprop\]) we get: $y_p(0)=1$ and $y_p(N)=x_{\max}$. The monotony of $h_f$ implies that $y_p(i-1) > y_p(i)$ for $i=1,2,..,N$. The above ansatz determines the map $f(x)$ in the interval $[0,x_{\max}]$. To find $f(x)$ in the remaining interval $[x_{\max},1]$ we take advantage of eq.(\[funx\]) and therefore have to invert the auxiliary function $h_f(x)$. It is straightforward to show that the expression (\[hfexp\]) can be inverted analytically leading to a closed form for $h_f^{-1}(x)$. The above defined ansatz for $h_{f}$ is in general piecewise smooth, i.e. smooth with the exception of a finite number of points.

In model II we use the following ansatz for the element $h_{f,i}$: $$\begin{aligned}
h_{f,i}(x)&=&y_p(i-1) - c_L(i) (x-x_p(i-1))^{\alpha_L(i)}~;~
x \in [x_p(i-1),x_s(i)) \label{hf2eq1}\\
h_{f,i}(x)&=&y_p(i) + c_R(i) (x_p(i)-x)^{\alpha_R(i)}~~~~;~~~~
x \in [x_s(i),x_p(i)] \label{hf2eq2}\end{aligned}$$ where we have introduced four new parameters $c_L(i),c_R(i),\alpha_R(i),x_s(i)$ and a new point $x_s(i) \in [x_p(i-1),x_p(i))$ for each $i=1,...,N$. Two of these parameters, namely $c_L(i)$ and $c_R(i)$, can be fixed demanding continuity of $h_f$ and 