  We study a class $\mathfrak X^H$ of signed Takagi-Landsberg functions with
Hurst parameter $H\in(0,1)$. We first show that the functions in $\mathfrak
X^H$ admit a linear $p^{\text{th}}$ variation along the sequence of dyadic
partitions of $[0,1]$, where $p=1/H$. The slope of the linear increase can be
represented as the $p^{\text{th}}$ absolute moment of the infinite Bernoulli
convolution with parameter $2^{H-1}$. The existence of a continuous
$p^{\text{th}}$ variation enables the use of the functions in $\mathfrak X^H$
as test integrators for higher-order pathwise It\^o calculus. Our next results
concern the maximum, the maximizers, and the modulus of continuity of the
classical Takagi-Landsberg function for all $0<H<1$. Then we identify the
uniform maximum, the uniform maximal oscillation, and a uniform modulus of
continuity for the class $\mathfrak X^H$.
