  We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is
of interest from two different perspectives. First, it is closely related to
sets of functions that have been studied as generalizations of the Takagi
function. Second, each function in $\mathscr{X}$ admits a linear pathwise
quadratic variation and can thus serve as an integrator in F\"ollmer's pathwise
It\=o calculus. We derive several uniform properties of the class
$\mathscr{X}$. For instance, we compute the overall pointwise maximum, the
uniform maximal oscillation, and the exact uniform modulus of continuity for
all functions in $\mathscr{X}$. Furthermore, we give an example of a pair
$x,y\in\mathscr{X}$ such that the quadratic variation of the sum $x+y$ does not
exist.
