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We study a class $\mathfrak X^H$ of signed Takagi-Landsberg functions withHurst parameter $H\in(0,1)$. We first show that the functions in $\mathfrakX^H$ admit a linear $p^{\text{th}}$ variation along the sequence of dyadicpartitions of $[0,1]$, where $p=1/H$. The slope of the linear increase can berepresented as the $p^{\text{th}}$ absolute moment of the infinite Bernoulliconvolution 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 resultsconcern the maximum, the maximizers, and the modulus of continuity of theclassical Takagi-Landsberg function for all $0<H<1$. Then we identify theuniform maximum, the uniform maximal oscillation, and a uniform modulus ofcontinuity for the class $\mathfrak X^H$.

On (signed) Takagi-Landsberg functions: $p^{\text{th}}$ variation, maximum, and modulus of continuity