The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity — a continuum — has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz’s famous apothegm natura non facit saltus — “nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit.

Artist: Giotto di Bondone (Colle di Vespignano, Mugello c. 1267 – Florence 1337)
Title: Coronation of the Virgin with Angels and Saints (Baroncelli Polyptych)
Date: after 1328; framed in Domenico Ghirlandaio’s workshop (end of XV century)
Dimensions: 186 x 342 x 27 cm
Material and tecnique: tempera and gold on wood; carved and gilded wood (frame)
Inscription: “OPUS MAGISTRI IOCTI”
Position: Basilica of Santa Croce, south aisle, Baroncelli chapel, altar

John L. Bell, "Continuity and Infinitesimals" in Stanford Encyclopedia of Philosophy, 2013.