In this paper I am going to deal with two very different kinds of continuity. One is of mathematical kind and it is familiar to every student of calculus; the other was named by Poincaré — not very appropriately, as we shall see — physical continuity (le continu physique). While the obvious contrast between these two different types of continuity is fairly well known, its deeper philosophical significance is rarely analyzed. This lack of interest in it is not accidental; it is due to the persistent influence of the intellectual tradition generated by the three centuries of classical science (1600–1900). We shall see that a more subtle epistemological approach together with the emergence of some new and quite unexpected problems in contemporary physics requires another fresh look at the contrast between both types of continuity and the way it was interpreted both by classical science and classical philosophy.