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Okamoto's one-parameter family of self-affine functions $F_a: [0,1]\to[0,1]$,where $0<a<1$, includes the continuous nowhere differentiable functions ofPerkins ($a=5/6$) and Bourbaki/Katsuura ($a=2/3$), as well as the Cantorfunction ($a=1/2$). The main purpose of this article is to characterize the setof points at which $F_a$ has an infinite derivative. We compute the Hausdorffdimension of this set for the case $a\leq 1/2$, and estimate it for $a>1/2$.For all $a$, we determine the Hausdorff dimension of the sets of points where:(i) $F_a'=0$; and (ii) $F_a$ has neither a finite nor an infinite derivative.The upper and lower densities of the digit $1$ in the ternary expansion of$x\in[0,1]$ play an important role in the analysis, as does the theory of$\beta$-expansions of real numbers.

The infinite derivatives of Okamoto's self-affine functions: an application of beta-expansions