What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§4. PRECISE DEFINITION OF CONTINUITY

In §1, Article 5 we stated what amounts to the following criterion for the continuity of a function: “A function f(x) is continuous at the point x = x₁ if, when x approaches x₁, the quantity f(x) approaches the value f(x₁) as a limit.” If we analyze this definition we see that it consists of two different requirements:

a) the limit a of f(x) must exist as x tends to x₁,

b) this limit a must be equal to the value f(x₁).

If in the limit definition of page 305 we set a = f(x₁), then the condition for continuity takes the following form: The function f(x) is continuous for the value x = x₁ if, corresponding to every positive number ε, no matter how small, there may be found a positive number δ (depending on ε ) such that

|f(x) – f(x₁)|< ε

for all x satisfying the inequality

| x – x₁ | < δ.

(The restriction x ≠ x₁ imposed in the limit definition is unnecessary here, since the inequality |f(x₁) – f(x₁)|< ε is automatically satisfied.)

Fig. 170. A function continuous at x = x₁.

Fig. 171. A function discontinuous at x = x₁.

As an example, let us check the continuity of the function f(x) = x³ a at the point x₁ = 0, say. We have

f(x₁) = 0³ = 0.

Now let us assign any positive value to ε, for example . Then we must show that by confining x to values sufficiently near x₁ = 0, the corresponding values of f(x) will not differ from 0 by more than i.e. will lie between and . We see immediately that this margin is not exceeded if we restrict x to values differing from X₁ = 0 by less than . In the same way we can replace , or whatever margin we desire; will always satisfy the requirement, since if

On the basis of the (ε,δ o)-definition of continuity one can show in a similar way that all polynomials, rational functions, and trigonometric functions are continuous, except for isolated values of x where the functions may become infinite.

In terms of the graph of a function u = f(x), the definition of continuity takes the following geometrical form. Choose any positive number ε and draw parallels to the x-axis at a height f(x₁) — ε and f(x₁) + ε above it. Then it must be possible to find a positive number δ such that the whole portion of the graph which lies within the vertical band of width 2δ about x₁ is also contained within the horizontal band of width 2ε aboutf(x₁). Figure 170 shows a function which is continuous at x₁, while Figure 171 shows a function which is not. In the latter case, no matter how narrow we make the vertical band about x₁, it will always include a portion of the graph that lies outside the horizontal band corresponding to the choice of ε.

If I assert that a given function u = f(x) is continuous for the value x = x₁, it means that I am prepared to fulfill the following contract with you. You may choose any positive number ε, as small as you please, but fixed. Then I must produce a positive number δ such that | x – x₁ | < δ implies | f(x) – f(x₁) | < ε. I do not contract to produce at the outset a number δ that will suffice for whatever ε you may subsequently choose; my choice of δ will depend on your choice of ε. If you can produce but one value ε for which I cannot provide a suitable δ, then my assertion is contradicted. Hence to prove that I can fulfill my contract in any concrete case of a function u = f (x), I usually construct an explicit positive function

δ = φ(ε),

defined for every positive number ε, for which I can prove that | x – x₁ | < δ implies always | f(x) – f(x₁) | < ε. In the case of the function u = f(x) = x³ at the value x₁ = 0, the function

Exercises: 1) Prove that sin x, cos x are continuous functions.

2) Prove the continuity of 1/(1 + x⁴) and of .

It should now be clear that the (ε, δ)-definition of continuity agrees with what might be called the observable facts concerning a function. As such it is in line with the general principle of modern science that sets up as the criterion for the usefulness of a concept or for the “scientific existence” of a phenomenon the possibility of its observation (at least in principle) or of its reduction to observable facts.