  A Brownian motion with drift is simply a process $V^{\eta}_t$ of the form
$V^{\eta}_t=B_{t}+\eta t$ where $B_{t}$ is a standard Brownian motion and
$\eta>0$ \footnote{The case $\eta<0$ is deducible by remarking
$V^{-\eta}(t)=-V^{\eta}(t)$.} In \cite{tanre2006range}, the authors considered
the drifted Brownian motion and studied the statistics of some related
sequences defined by certain stopping times. In particular, they provided the
law of the range $R_{t}(V^{\eta})$ of $V^{\eta}$ as well as its first range
process $\theta_{V^{\eta}}(a)$. In particular, they investigated the asymptotic
comportment of $R_{t}(V^{\eta})$ and $\theta_{V^{\eta}}(a)$. They proved that
if $V_{t}^{\eta}$ is a Brownian motion with a positive drift $\eta$ then its
range $R_{t}(V^{\eta})=\sup_{0\leq s\leq t}V_{t}^{\eta}-\inf_{0\leq s\leq
t}V_{t}^{\eta}$ is asymptotically equivalent to $\eta t$. In other words
  \begin{equation}
\frac{R_{t}(V^{\eta})}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}\eta.\label{range}
\end{equation} In this paper, we show that (\ref{range}) follows from a
striking deterministic property. More precisely, we show that the long run
behavior of the range of a deterministic continuous function is obtainable
straightaway from that of the function itself. Our result can be deemed as the
continuous version of a similar one appeared in \cite{mgrw}.
