  We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$
and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the
parameter $t$. Each function of the class is defined by a certain pair of two
square matrices of order two. The class includes the Lebesgue singular
functions and other singular functions. Our approach to the Takagi function is
similar to Hata and Yamaguti. The class of partial derivatives
$\partial_{t}^{k} F(t, x)$ includes the original Takagi function and some
generalizations. We consider real-analytic properties of $\partial_{t}^{k} F(t,
x)$ as a function of $x$, specifically, differentiability, the Hausdorff
dimension of the graph, the asymptotic around dyadic rationals, variation, a
question of local monotonicity and a modulus of continuity. Our results are
extensions of some results for the original Takagi function and some
generalizations.
