  This paper gives the pointwise H\"older (or multifractal) spectrum of
continuous functions on the interval $[0,1]$ whose graph is the attractor of an
iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$.
These functions satisfy a functional equation of the form $\phi(a_k x+b_k)=c_k
x+d_k\phi(x)+e_k$, for $k=1,2,\dots,r$ and $x\in[0,1]$. They include the Takagi
function, the Riesz-Nagy singular functions, Okamoto's functions, and many
other well-known examples. It is shown that the multifractal spectrum of $\phi$
is given by the multifractal formalism when $|d_k|\geq |a_k|$ for at least one
$k$, but the multifractal formalism may fail otherwise, depending on the
relationship between the shear parameters $c_k$ and the other parameters. In
the special case when $a_k>0$ for every $k$, an exact expression is derived for
the pointwise H\"older exponent at any point. These results extend recent work
by the author [Adv. Math. 328 (2018), 1-39] and S. Dubuc [Expo. Math. 36
(2018), 119-142].
