A Tour of the Calculus by David Berlinski

A Tour of the Calculus by David Berlinski

Author:David Berlinski [Berlinski, David]
Language: eng
Format: epub
ISBN: 978-0-307-78973-0
Publisher: Knopf Doubleday Publishing Group
Published: 2011-04-27T05:00:00+00:00


4 Suppose that f(x) is −1 whenever x is greater than or equal to 0 but less than 2, i.e., 0 ≤ x < 2; but that it is 1 whenever x is greater than or equal to 2 but less than or equal to 4, i.e., 2 ≤ x ≤ 4. This strange function jumps over the x-axis at 2 itself, violating the conclusion of the intermediate value theorem. But, of course, f is not continuous at 2 either. The picture shows what is at issue:

5 Suppose the function f(x) has the value x2 if x is less than 1, but if x is greater than or equal to 1, f(x) drops to 0, where it forever after remains. This abject function is bounded on the closed interval [0, 1] (by 1 itself), but it does not take its maximal value on this interval. The natural thought that f(1) must be greater than any other value of f comes to grief on the realization that f(1) is 0 and so less than the other values clambering upward. Again, a discontinuity at a single point (1, in fact) destroys the conclusion of the theorem. And again a picture shows the moral of this message better than the message shows the moral:



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