  Let $\Omega$ be a set of positive integers and let $f:\Omega \rightarrow
\Omega$ be an arithmetic function. Let $V = (v_i)_{i=1}^n$ be a finite sequence
of positive integers. An integer $m \in \Omega$ has
\textit{increasing-decreasing pattern} $V$ with respect to $f$ if, for all odd
integers $i \in \{1,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) < f^{v_1+
\cdots + v_{i-1}+1}(m) < \cdots < f^{v_1+ \cdots + v_{i-1}+v_{i}}(m) \] and,
for all even integers $i \in \{2,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m)
> f^{v_1+ \cdots +v_{i-1}+1}(m) > \cdots > f^{v_1+ \cdots +v_{i-1}+v_i}(m). \]
The arithmetic function $f$ is \textit{wildly increasing-decreasing} if, for
every finite sequence $V$ of positive integers, there exists an integer $m \in
\Omega$ such that $m$ has increasing-decreasing pattern $V$ with respect to
$f$. This paper gives a proof that the Syracuse function is wildly
increasing-decreasing.
