  Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called "little
lip" function $\operatorname{lip} f$ is defined as follows: \begin{equation*}
\operatorname{lip}
  f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r}
\frac{|f(y)-f(x)|}{r}. \end{equation*} We show that if $f$ is continuous on
$\mathbb{R}$, then the set where $\operatorname{lip} f$ is infinite is a
countable union of a countable intersection of closed sets (that is an
$F_{\sigma \delta}$ set). On the other hand, given a countable union of closed
sets $E$, we construct a continuous function $f$ such that $\operatorname{lip}
f$ is infinite exactly on $E$. A further result is that for the typical
continuous function $f$ on the real line $\operatorname{lip} f$ vanishes almost
everywhere.
