100 CHAPTER 6. CONTINUOUS FUNCTIONS

Weierstrass2 is the precise way to exclude the sort of behavior described above and allstatements about continuous functions must ultimately rest on this definition from nowon or something which is equivalent to it. I am going to present this in the context offunctions which are defined on D( f ) ⊆ Fp having values in Fq where p,q are positiveintegers because it is no harder. However, in most of the applications in this book, D( f )will be in R or C.

Definition 6.0.1 A function f : D( f ) ⊆ Fp → Fq is continuous at x ∈ D( f ) if foreach ε > 0 there exists δ > 0 such that whenever y ∈ D( f ) and ∥y− x∥< δ it follows that∥ f (x)− f (y)∥< ε. A function f is continuous if it is continuous at every point of D( f ) .

If f has values in Fp, it is of the form x→ ( f1 (x) , · · · , fp (x)) where the fi are realvalued functions.

In sloppy English this definition says roughly the following: A function f is continuousat x when it is possible to make f (y) as close as desired to f (x) provided y is taken closeenough to x. In fact this statement in words is pretty much the way Cauchy described it.The completely rigorous definition above is usually ascribed to Weierstrass.

If you are like me, you may find the following equivalent description of continuityeasier to remember and use. I don’t have a very good reason why this is so, but it seems tobe the case, at least for many people. I will use either definition whenever convenient.

Theorem 6.0.2 A function f is continuous if and only if whenever xn → x withxn,x ∈ D( f ) , it follows that f (xn)→ f (x). In words, convergent sequences get takento convergent sequences.

Proof: ⇒ Suppose xn → x as described. I need to verify that f (xn)→ f (x) . I knowthat for any ε > 0 there exists a suitable δ such that the conditions of continuity hold. I alsoknow that, since xn→ x, eventually, for all n large enough, ∥xn− x∥< δ . Therefore, for alln large enough, ∥ f (x)− f (xn)∥ < ε , but this is the definition of what it means to say thatf (xn)→ f (x).⇐ Suppose the sequence condition holds. Why is f continuous at x? If it isn’t, then

there exists ε > 0 for which there is no suitable definition from the definition of continuity.Hence 1/n is not a suitable δ for this ε . It follows that there exists xn such that ∥xn− x∥<1/n and yet ∥ f (xn)− f (x)∥ ≥ ε. But then xn→ x and f (xn)↛ f (x) where the symbol↛indicates that f (xn) does not converge to f (x). Hence f must be continuous at x after all.

This definition or its equivalent formulation rules out the sorts of graphs drawn above.Consider the second nonremovable discontinuity. The removable discontinuity case is

similar. You could let xn→ x0 where each xn < x0 and the limit of f (xn) will fill in the holeat the bottom of the graph although the actual value of the function at f (x0) is larger. Thusf (xn)↛ f (x0) so f is not continuous at x0.

2Wilhelm Theodor Weierstrass 1815-1897 brought calculus to essentially the state it is in now. When he wasa secondary school teacher, he wrote a paper which was so profound that he was granted a doctor’s degree. Hemade fundamental contributions to partial differential equations, complex analysis, calculus of variations, andmany other topics. He also discovered some pathological examples such as nowhere differentiable continuousfunctions. Cauchy and Bolzano were the first to use the ε δ definition presented here but this rigorous definitionis associated more with Weierstrass. Cauchy clung to the notion of infinitesimals and Bolzano’s work was notreadily available. The need for rigor in the subject of calculus was only realized over a long period of time andthis definition is part of a trend which went on during the nineteenth century to define exactly what was meant.