  This paper studies a large class of continuous functions
$f:[0,1]\to\mathbb{R}^d$ whose range is the attractor of an iterated function
system $\{S_1,\dots,S_{m}\}$ consisting of similitudes. This class includes
such classical examples as P\'olya's space-filling curves, the Riesz-Nagy
singular functions and Okamoto's functions. The differentiability of $f$ is
completely classified in terms of the contraction ratios of the maps
$S_1,\dots,S_{m}$. Generalizing results of Lax (1973) and Okamoto (2006), it is
shown that either (i) $f$ is nowhere differentiable; (ii) $f$ is
non-differentiable almost everywhere but with uncountably many exceptions; or
(iii) $f$ is differentiable almost everywhere but with uncountably many
exceptions. The Hausdorff dimension of the exceptional sets in cases (ii) and
(iii) above is calculated, and more generally, the complete multifractal
spectrum of $f$ is determined.
