Lipschitz Continuity

Abstract

When we graph a function f (x) of a rational variable x, we make a leap of faith and assume that the function values f (x) vary “smoothly” or “continuously” between the sample points x, so that we can draw the graph of the function without lifting the pen. In particular, we assume that the function value f (x) does not make unknown sudden jumps for some values of x. We thus assume that the function value f (x) changes by a small amount if we change x by a small amount. A basic problem in Calculus is to measure how much the function values f (x) may change when x changes, that is, to measure the “degree of continuity” of a function. In this chapter, we approach this basic problem using the concept of Lipschitz continuity, which plays a basic role in the version of Calculus presented in this book.

Calculus required continuity, and continuity was supposed to require the infinitely little, but nobody could discover what the infinitely little might be. (Russell)