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 . We may restrict x to a fixed interval |x| ≤ M, where M, is an arbitrarily selected number. Writing
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 .
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.