Graphs and Continuity—in which we arrange a marriage between Intuition and Rigour

Abstract

The mathematical idea of continuity is analogous to, but not the same as, the intuitive idea of continuity which we associate with time, space or motion. We think of time as unbroken, of space as smooth with no holes and of motion as uninterrupted. The mathematician, perverse as ever, seeks to redefine this comfortable vague notion of continuity by a more useful, more precise but more troublesome definition. The real, line, which we have taken such pains to define, is deemed to be continuous. Recall that we required the completeness axiom to plug the imperceptible gaps. In this chapter we shall be mainly concerned with the notion of a ‘continuous function’. The definition of this concept is necessarily precise but it accords, most of the time, with our notion of an unbroken curve. The current theory of mathematical continuity is an abstract logical edifice which may or may not describe the way space actually is. So far, mathematicians have been able to resolve any unexpected quirks of the rigorously defined concept of a continuous function more or less to everyone’s satisfaction. One of the founders of analysis, a Catholic priest, Bernhard Bolzano (1781–1848 ), when analysing the paradoxes of the infinite, was driven to define various intuitively obvious mathematical concepts such as continuity.

‘It then only remained to discover … a real definition of the essence of continuity. I succeeded 24 November 1858, and a few days afterward I communicated the results to my dear friend Durege with whom I had a long and lively discussion.’

Richard Dedekind