  We investigate the generalization of the mistake-bound model to continuous
real-valued single variable functions. Let $\mathcal{F}_q$ be the class of
absolutely continuous functions $f: [0, 1] \rightarrow \mathbb{R}$ with
$||f'||_q \le 1$, and define $opt_p(\mathcal{F}_q)$ as the best possible bound
on the worst-case sum of the $p^{th}$ powers of the absolute prediction errors
over any number of trials. Kimber and Long (Theoretical Computer Science, 1995)
proved for $q \ge 2$ that $opt_p(\mathcal{F}_q) = 1$ when $p \ge 2$ and
$opt_p(\mathcal{F}_q) = \infty$ when $p = 1$. For $1 < p < 2$ with $p =
1+\epsilon$, the only known bound was $opt_p(\mathcal{F}_{q}) =
O(\epsilon^{-1})$ from the same paper. We show for all $\epsilon \in (0, 1)$
and $q \ge 2$ that $opt_{1+\epsilon}(\mathcal{F}_q) =
\Theta(\epsilon^{-\frac{1}{2}})$, where the constants in the bound do not
depend on $q$. We also show that $opt_{1+\epsilon}(\mathcal{F}_{\infty}) =
\Theta(\epsilon^{-\frac{1}{2}})$.
