15 Real Analysis

Real analysis is the mathematical subject concerned in its beginnings with the real num-

bers and with functions on the real numbers. The subject refines and extends many ideas

usually first encountered in elementary calculus, including differentiation and integration,

but often generalizes them to higher realms, to abstract spaces with infinitely many di-

mensions or exotic measures. In this chapter, however, we shall remain a little closer to

ground, concentrating on the real numbers and continuous real-valued functions on the real

numbers.

15.1 Definition of continuity

Perhaps the reader has studied calculus and has experience with the real numbers and

with continuous real-valued functions. What does it mean, precisely, for a function to be

continuous? In elementary calculus, one is sometimes content at first with an informal

concept of continuity. For example, perhaps in high school one might have heard the

following account:

A function is continuous if you can draw it without lifting your pencil.

continuous not continuous

That phrase does suggest that a jump discontinuity, as in the red function at the right, should

make a function discontinuous, since indeed you would have to lift your pencil to draw it.

But is the meaning of the statement precise enough to serve adequately in a mathematical

argument? I do not think so, and in the end I take this phrase as a suggestive metaphor

rather than as a definition.