4.2: Some General Theorems on Limits and Continuity

I. In §1 we gave the so-called "" definition of continuity. Now we present another (equivalent) formulation, known as the sequential one. Roughly, it states that is continuous iff it carries convergent sequences into convergent "image sequences" More precisely, we have the following theorem.

Theorem (sequential criterion of continuity).

(i) A function

is continuous at a point iff for every sequence such that in we have in In symbols,

(ii) Similarly, a point is a limit of at iff

Note that in (2') we consider only sequences of terms other than .

Proof

We first prove (ii). Suppose is a limit of at i.e. (see §1),

Thus, given there is (henceforth fixed) such that

We want to deduce (2'). Thus we fix any sequence

Then

and contains all but finitely many Then these satisfy the conditions stated in (3). Hence for all but finitely many As is arbitrary, this implies (by the definition of as is required in (2'). Thus (2) (2').

Conversely, suppose (2) fails, i.e., its negation holds. (See the rules for forming negations of such formulas in Chapter 1, §§1-3.) Thus

by the rules for quantifiers. We fix an satisfying (4), and let

By (4), for each there is such that

and

We fix these As and we obtain a sequence

Also, as we have and hence . On the other hand, by (6), the image sequence canverge to (why?), i.e., (2') fails. Thus we see that (2') fails or holds accordingly as (2) does.

This proves assertion (ii). Now, by setting in (2) and (2'), we also obtain the first clause of the theorem, as to continuity.

Note 1. The theorem also applies to relative limits and continuity over a path (just replace by in the proof), as well as to the cases and in (for can be treated as a metric space; see the end of Chapter 3, §11).

If the range space is complete (Chapter 3, §17), then the image sequences converge iff they are Cauchy. This leads to the following corollary.

Corollary 1. Let be complete, such as Let a map with and a point be given. Then for to have a limit at it suffices that be Cauchy in whenever and in

Indeed, as noted above, all such converge. Thus it only remains to show that they tend to one and the same limit as is required in part (ii) of Theorem 1. We leave this as an exercise (Problem 1 below).

Theorem (Cauchy criterion for functions).

With the assumptions of Corollary 1, the function has a limit at iff for each there is such that

In symbols,

Proof

Assume (7). To show that has a limit at we use Corollary 1. Thus we take any sequence

and show that is Cauchy, i.e.,

To do this, fix an arbitrary By (7), we have

for some Now as there is such that

As we even have Hence by (7'),

i.e., is Cauchy, as required in Corollary 1, and so has a limit at . This shows that (7) implies the existence of that limit.

The easy converse proof is left to the reader. (See Problem 2.)

II. Composite Functions. The composite of two functions

denoted

is by definition a map of into given by

Our next theorem states, roughly, that is continuous if and are. We shall use Theorem 1 to prove it.

Theorem

Let and be metric spaces. If a function is continuous at a point and if is continuous at the point then the composite function is continuous at .

Proof

The domain of is So take any sequence

As is continuous at formula (1') yields where is in Hence, as is continuous at we have

and this holds for any with Thus satisfies condition (1') and is continuous at

Caution: The fact that

does not imply

(see Problem 3 for counterexamples).

Indeed, if and we obtain, as before, but not Thus we cannot re-apply formula (2') to obtain since (2') requires that The argument still works if is continuous at (then (1') applies) or if never equals then . It even suffices that for in some deleted globe about (see §1, Note 4). Hence we obtain the following corollary.

Corollary 2. With the notation of Theorem 3, suppose

Then

provided, however, that

(i) is continuous at or

(ii) for in some deleted globe about or

(iii) is one to one, at least when restricted to some .

Indeed, (i) and (ii) suffice, as was explained above. Thus assume (iii). Then can take the value at most once, say, at some point

As let

Then so on and case (iii) reduces to (ii).

We now show how to apply Corollary 2.

Note 2. Suppose we know that

Using this fact, we often pass to another variable setting where is such that for some We shall say that the substitution (or "change of variable") is admissible if one of the conditions (i), (ii), or (iii) of Corollary 2 holds. Then by Corollary 2,

(yielding the second limit).

Example

(A) Let

Then

For a proof, let be the integral part of Then for ,

As tends to over integers, and by rules for sequences,

with as in Chapter 3, §15. Similarly one shows that also

Thus (8) implies that also (see Problem 6 below).

Remark. Here we used Corollary 2(ii) with

The substitution is admissible since never equals its limit, thus satisfying Corollary 2(ii).

(B) Quite similarly, one shows that also

See Problem 5.

(C) In Examples and we now substitute This is admissible by Corollary 2(ii) since the dependence between and is one to one. Then

Thus and yield

Hence by Corollary 3 of §1, we obtain