246 Chapter 5 • Continuous Functions
Statement #1 is stronger than Statement #2; that is, #1 => #2 but #2 =fo #l.
We give an example in which #2 is true but #1 is false (see also Exercise 1).
Example 5.3.1 Consider the function f : JR---+ JR given by f(x) = X[o,11(x),
the "characteristic function"^9 of the interval [O, l].
yxFigure 5.6In this example,- f : JR ---+ JR is not continuous on [O, 1]; it is not continuous at 0 since
lim f(x) =f. lim f(x).
x->O+ x->O- - f : [O, 1] ---+ JR is continuous on [O, 1 J, since in determining its continuity,
no x outside [O, 1] may be used. f is continuous at every a E [O, 1] since
Ve> 0, :lb> 0 3 Vx E [O, 1], 0 <Ix -al< 5 => lf(x) - ll = 0 < c. •
Because Statement #1 is stronger than Statement #2, we prefer to use #1
in conclusions of theorems and #2 in hypotheses, whenever possible. Remem-
ber, a theorem is strongest when its hypotheses are as weak as we can make
them and its conclusion is as strong as we can make it.
So far, the notation we are using does not distinguish between the function
symbol f used for f : A ---+ JR and the symbol f used for f : V(f) ---+ R
Occasionally, it is useful to have two different symbols to distinguish these two
different functions.
Definition 5.3.2 Suppose f : V(f) ---+JR and A s;;; V(f). We define the func-
tion !IA (called "f restricted to A") as follows:
(a) The domain of !IA is A;
(b) Vx EA, flA(x) = f(x).- Characteristic functions are defined in Exercise 5.2.5.