  In this paper we study the family of $\alpha$-Farey-Minkowski functions
$\theta_\alpha$, for an arbitrary countable partition $\alpha$ of the unit
interval with atoms which accumulate only at the origin, which are the
conjugating homeomorphisms between each of the $\alpha$-Farey systems and the
tent map. We first show that each function $\theta_\alpha$ is singular with
respect to the Lebesgue measure and then demonstrate that the unit interval can
be written as the disjoint union of the following three sets:
$\Theta_0:={x\in\U:\theta_\alpha'(x)=0},
\Theta_\infty:={x\in\U:\theta_\alpha'(x)=\infty} and
\Theta_\sim:=\U\setminus(\Theta_0\cup\Theta_\infty)$. The main result is that
[\dim_{\mathrm{H}}(\Theta_\infty)=\dim_{\mathrm{H}}(\Theta_\sim)=\sigma_\alpha(\log2)<\dim_{\mathrm{H}}(\Theta_0)=1,]
where $\sigma_\alpha(\log2)$ is the Hausdorff dimension of the level set ${x\in
\U:\Lambda(F_\alpha, x)=s}$, where $\Lambda(F_\alpha, x)$ is the Lyapunov
exponent of the map $F_\alpha$ at the point $x$. The proof of the theorem
employs the multifractal formalism for $\alpha$-Farey systems.
