Abstract
Given a function f : R -> R, the so-called "little lip" function lip f is defined as follows: lip f(x) = lim inf(r -> 0) sup(|x - y| <= r) |f(y) - f(x)| / r. We show that if f is continuous on R, then the set where lip f is infinite is a countable union of countable intersections of closed sets (that is, an F-sigma delta set).
On the other hand, given a countable union E of closed sets, we construct a continuous function f such that lip f is infinite exactly on E. A further result is that, for a typical continuous function f on the real line, lip f vanishes almost everywhere.