EXAMPLE ON CONTINUITY - MORE EXAMPLES ON LIMITS AND CONTINUITY - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

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

SUPPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINUITY

§2. EXAMPLE ON CONTINUITY

To give a precise proof of the continuity of a function requires the explicit verification of the definition of page 310. Sometimes this is a lengthy procedure, and therefore it is fortunate that, as we shall see in Chapter VIII, continuity is a consequence of differentiability. Since the latter will be established systematically for all elementary functions, we may follow the usual course of omitting tedious individual proofs of continuity. But as a further illustration of the general definition we shall analyze one further example, the function image. We may restrict x to a fixed interval |x| ≤ M, where M, is an arbitrarily selected number. Writing

image

we find for |x | ≤ M, and |x1 | ≤ M,

|f (x1)f(x)| ≤ |x – x1||x – x1| ≤ |x – x1|·2M.

Hence it is clear that the difference on the left side will be smaller than any positive number ε if only image.

It should be noted that we are being quite generous in our appraisals. For large values of x and x1 the reader will easily see that a much larger δ would suffice.