A Brownian motion with drift is simply a process $V^η_t$ of the form $V^η_t=B_{t}+ηt$ where $B_{t}$ is a standard Brownian motion and $η>0$ \footnote{The case $η<0$ is deducible by remarking $V^{-η}(t)=-V^η(t)$.} 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 $R_{t}(V^η)$ of $V^η$ as well as its first range process $θ_{V^η}(a)$. In particular, they investigated the asymptotic comportment of $R_{t}(V^η)$ and $θ_{V^η}(a)$. They proved that if $V_{t}^η$ is a Brownian motion with a positive drift $η$ then its range $R_{t}(V^η)=\sup_{0\leq s\leq t}V_{t}^η-\inf_{0\leq s\leq t}V_{t}^η$ is asymptotically equivalent to $ηt$. In other words \begin{equation} \frac{R_{t}(V^η)}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}η.\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}.