Why Read This Book? - Index of

7.6 Implications of Continuity 231
An immediate consequence of Exercise 7.3.2 is the following.
Theorem 7.5.20 A function f is continuous at a if and only if it is both left and
right continuous at a.
In defining continuity on a set, Definition 7.5.17 stipulated that the set must
be open. This allowed for a definition of continuity on (a, b), but not on [a, b].
One-sided continuity now allows us to define continuity on [a, b] in such a way
that we do not concern ourselves with how f behaves or whether it even exists
outside of [a, b].
Definition 7.5.21 A function f is said to be continuous on [a, b] provided the
following hold:
1. f is continuous on (a, b),
2. f is right continuous at a,
3. f is left continuous at b.
Definition 7.5.21 can be naturally adapted to apply to the following example
by omitting stipulation 3.
Example 7.5.22 f(x) = √ x is continuous on [0, +∞). From Exercise 7.5.18,
f is continuous at all positive real numbers, and by Exercise 7.3.5, it is right
continuous at 0.
7.6 Implications of Continuity
Continuity has many implications, and in this section we look at three of them.
The second of these results is actually logically equivalent to continuity.
7.6.1 The Intermediate Value Theorem
The imagery that we can sketch a continuous function without picking up the
pencil makes this first result seem plausible. It says that a continuous function
cannot be negative at one point and positive somewhere else without crossing the
x-axis somewhere between the two points. One case of this is stated in the next
exercise. To prove it, imagine standing on the x-axis at a and looking up the x-axis.
There is a natural subset of the real numbers that is both non-empty and bounded
from above by b to which the LUB property can apply.
EXERCISE 7.6.1 Suppose a