The purpose of this chapter is to bring out appropriate mathematical concepts to express whether a subset of ℝ (to begin with) is small or large. The notion of smallness that we would like to define is subject to three conditions.

1. (1) Heredity: any subset of a small set must also be small.

2. (2) Stability under countable union (any countable union of small sets is also small).

3. (3) No interval [a, b] (with a < b) is small.

A subset of ℝ will be large if its complement is small. If P(x) is an assertion depending on a real number x, we say that P is generic (or typical) if P(x) is true for x belonging to a large subset of ℝ;.

Here, among others, are three possible points of view.

1. • Cardinality: the small sets are those that are finite or countable.

2. • Measure: the small sets are those that are negligible in the sense of Lebesgue.

3. • Category: the small sets are those that are of first category in the sense of Baire.

In what follows, we will leave the first point of view aside, in order to compare the other two notions in a specific situation: the study of the points of continuity of derivative functions. For all this chapter, a good reference is [32].