  We give a formula for $f(\eta)$, where $f :\mathbb C \to \mathbb C$ is a
continuously differentiable function satisfying $f(\bar z) = \overline{f(z)}$,
and $\eta$ is a dual quaternion. Note this formula is straightforward or well
known if $\eta$ is merely a dual number or a quaternion. If one is willing to
prove the result only when $f$ is a polynomial, then the methods of this paper
are elementary.
